My dad does this and it turns out great. He takes a roasting pan (with the metal grate in the bottom so the meat doesn't rest on a flat surface) places a large un-cut piece of meat on the grate and then covers that with paper towels. He keeps it on the bottom rack of the fridge (not sure if that' just the shelf with the most room or if he has a reason) and lets it age for a week or so. Obviously if it gets funky don't eat it, but it should age nicely in the dry, cool fridge and break down just enough to be tender!

Alton Brown has done a Good Eats episode about this - it would probably be worth checking out if you're looking to do it at home. The FN recipe page for one of the recipes has a brief rundown but if you can catch the episode (Family Roast) it would likely help.
http://www.foodnetwork.com/recipes/alton-brown/dry-aged-standing-rib-roast-with-sage-jus-recipe/index.html

I've done this with steaks. I put them in a small baking dish, resting on the small side of the meat, exposing as much surface area of the meat as possible to the air. Then I turn my refrigerator's plastic egg holder upside down on top of the baking dish, so that there's enough space for air to circulate underneath. I've aged them this way for a week and it worked well. Once I tried lightly salting the meat before I started the aging and that was not good, it turned out too dry and chewy. You could also do this with some old tupperware, if you cut some holes in it for ventilation.