The first thing I noticed was that you did not have a hood over the range. If venting outside is impossible, most hood units have a recirculating feature with washable (or replaceable) mesh filters that collect much of the grease before it settles elsewhere, like the upper regions of your cabinets. You can ask your landlord to install one if you rent!
I'm a kitchen designer and am often surprised how often proper ventilation is set aside as a low priority. That alone will help with the majority of the grease. The dust, however, is another matter.....Best of luck!

I think picnic baskets look really cute above above kitchen cabinets, they are food themed and have a lid to protect their contents.

Also you might consider hanging some curtains above the cabinets. Use some aircraft cable and some eye hooks screwed into your walls or ceiling. Then you can put your stuff inside any old cardboard box or plastic bin and no one will see it. I think this would really cute in your kitchen since it's such a tight space. It would add some color to your kitchen and would hide some of the 'visual clutter' you'd get with regular containers.
If you use a cotton fabric you can even take it down and wash it once in a while.

I notice you have a lot of spare space over the stove. If installing a hood isn't an option, perhaps putting free-standing shelves over the stove, or a free-standing cupboard (like a medicine cabinet, but for spices perhaps?) would make things more accessible.

I've done this in my current kitchen, and it makes an incredible amount of difference in storage space.

In addition, it means that the things over the stove (which get dirty more quickly) are the ones that are used more frequently (in my case, pots containing sugar, flour, oatmeal, etc) and these can be wiped down more frequently. This frees up cupboard space for the things that otherwise wind up getting dusty and gross on the top of the cupboards.

This won't help you, Helen, but here's what I did on my about-to-be-completed house: I wanted cabinets all the way to the ceiling, but I thought solid doors would look too monumental. So there are two cabinets in each bank of wall cabinets. One goes to within about 15"-18" of the ceiling. Above it is a cabinet with glass doors. So rarely-used/pretty stuff can go up top, but but still be protected behind closed doors.

What about some under-the-cabinet ideas, like a knife block, appliances, etc., that would give you more counter space between the stove and sink. I also use large baskets above my cabinets - but I also outfitted the closest closet to the kitchen as a pantry, with some steel shelving and simple nails on the side walls to hang pots, etc.

http://www.amazon.com/s/ref=nb_sb_ss_i_3_16?url=search-alias%3Dgarden&field-keywords=under cabinet knife block&sprefix=under cabinet kn

We have the same issue. I visit friends who have lovely decorative objects above their cupboards. We have cases of canning jars, cookbooks, lunchboxes, the salad spinner (awesome!) and other various seldom-used items. I'm not interested in hiding all the stuff, I want to know what the best cleaner is for the dusty grease that accumulates on the tops of the cabinets. And I did not know you can get a hood that doesn't need to be ventilated to the outside! Looking into that!

This is basically my kitchen. Two bits of advice. Hang pans in the space over the stove. Ikea sells a rack that is almost flat against the wall and is perfect for skillets. Put the skillets you use most often there, and you will use them often enough that it won't matter that they get greasy.

As far as the space over the cabinets, if you don't have a stove hood, it's very hard to use it without everything getting greasy. (I assume this is a rental and you're stuck without a hood.) Several people here have suggested using baskets, but the problem is that the baskets will get greasy, even if what's inside stays clean. Yuck. My only advice is to put things up there that are washable, and run them through the dishwasher every couple of months. Roasting pans are good - they're big so they take up a lot of space in the cabinet, but they're usually metal and easily washable. If you must put things up there, try to keep them away from the stove and push them back a bit - they stay pretty clean that way. (However, that makes them harder to reach.) Or wrap the items in towels that you can throw in the washer (or paper towels if you don't mind a little waste.)

I also have a corridor kitchen with the same space issues.

All I did is cover all my rarely used items that were on top of the cabinets with plastic shopping bags to prevent dust. It looks pretty crap, but is very effective.

If you want a more aesthetically pleasing solution, a friend of mine covered everything on top of her cabinet with off cuts of pretty fabric. She would just kind of drape them over the appliances and tuck the ends under.

This is not an issue that needs to involve much, if any spending. I wouldn't bother with pretty boxes and baskets - they'll just get wrecked anyway.

I've filled the space above my cabinets with jardeniers, an ArtDeco cut glass vase, brightly decorated gift boxes with ribbons, and an African ebony statue. Really, turn the space into an art collection and make the kitchen an unique expression of your style.

I have above the cabinet space too, and our kitchen is very tiny so I really wanted to use it for something good too. Looks to me like you've got a good foot of space up there!

Since I've only got about 8 inches, what I've ended up doing is using canning jars of various sizes to store my bulk bought beans, rice, popping corn, bay leaves, whatever. The jars are easy to de-grease as long as you can do it like once a month (otherwise, the dish washer suggestion is genius if you have one!) This has made room inside the cabinets for the less appealing stuff that I could put up there. Between the canning jars, I've got a couple cuttings of a plant growin' and that looks nice too. Little vines reaching down livens the space up a lot.

But I totally second the idea to hang a rack or shelves over the stove. I did that in my last apartment and put all my vintage glasses, dinnerware, and kitchen collections out on display while i hid all the wonky stuff in the one cabinet I had.

Good luck!

That's a though one! The best I can come up with would be to find some containers/baskets that fit well up there. The ikea baskets that fit in the billy bookcase come to mind. If the baskets are open a towel inside but over your stuff to help with the dust. You may have to DIY rehab whatever you pick to make it look better. I spray painted some wicker storage containers and they came out cute.

I would put some containers and then add a few select decorative accessories.

Helen here - thanks for all the suggestions, everybody! Yes, I probably should have mentioned that this is a rental, and smack in the middle of a very large/tall low-income building at that (fiancée is still a student and we are within walking distance of school/work), so I am not sure if they would install a range hood at all. They respond very well to broken things, but I've never tried asking for something new.

We also have mostly concrete walls and ceilings, so it's been very difficult to mount things in some places. However, I just checked the wall right around the stove and, happily, it is drywall! So, Ros and toberead - I will definitely look into options for using that space better. Thanks to gifts of things like a dutch oven and cast iron skillet, I'm thinking that shelves will be the best way to go so that I don't have to put those heavy (and pretty!) items too far away from the stove. Hopefully I can find shelves that easily wipe clean, since they will still get greasy.

So, back to that funny space... I had thought about boxes/bins/baskets, and still do, but I don't want those to get disgusting nor do I want to be climbing my cabinets every day to unbox something. I also wondered if I could hang curtains or even somehow install doors of some kind, but the concrete ceiling makes that impossible. Pieces of cloth sounds like a good idea, especially since it can be laundered - next time I'm somewhere with fabric leftovers, I'll take a look. Or wait until we go to IKEA for shelves!

Finally, mdorothy - your cabinet solution sounds exactly like what I would want if I ever get to design my own kitchen (along with, of course, a range hood).

What I have up there: box of all kitchen-related manuals, box of cookie cutters/pastry bags (which usually only come out around xmas), extra CO2 canisters for my Sodastream, my cast iron dutch oven (the Le Creuset stays in the cabinet, but the cast iron can benefit from the kitchen grease), and an 8-pack of paper towels (it's in plastic so no dust issues there). Is it pretty? No, not really. But it's functional.

I had the same problem. I went out and found three short, long cabinets that ar very similar to my cabinets and just sat them up there. Works great for those heirloom, non-micro safe dinerware sets. BTW, anything that sits for a month gets at least a rinse before using.

DoubleH:
You can hang things in a concrete ceiling. It is not difficult. You need a hammer drill and a masonry bit. After you get those two things, it works just like using a regular drill with a regular bit in a regular ceiling. You will need to get concrete anchors or tapcons (depending how much weight they are bearing) for the screws that go into the ceiling. These are all simple things that anyone at a hardware store can easily help you with. We learned all this after moving into our condo with concrete ceiling and cinder block walls.

I think hanging a curtain is a perfect idea, since it will be washable and won't get dusty-greasy like boxes would.

Your landlord probably would not want to go to the trouble of installing a venting hood in a multi-story building that didn't already have a pipe set up to go outside, but they might be willing to go for a recirculating hood - it never hurts to ask!

I use these bins from IKEA, but it looks like they no longer sell them (I did a search for "galvanized" and for the "SKRIN" name w/no luck). However, they're made of galvanized steel, come with lids, and are super-easy to clean.

I imagine you can find something similar in the lawn/garden center (say, planter boxes?) or even find a plastic version of the same style bin. Even though my boxes have lids, I like the idea others have suggested about placing a cloth/dish towel on top of the items inside.

If you don't want to/can't drill into your ceiling, you might be able to hang a curtain with some Velcro. You can get the kind with self-stick adhesive on the back (my mother hung a curtain in out bathroom like this) the glue it held up to the moisture pretty well. Just make sure the ceiling is clean before you put up the Velcro or it won't stay up.

I collected a bunch of wine boxes and have been using those to store things above my cabinet. At least that way it's a little more attractive (and easier to take things down). Just measure your space before you go hunting for boxes - they come in a wide range of heights!