There's some debate whether or not oriental rugs belong in the kitchen, but rag rugs are a different story. These colorful throw rugs are casual, easy to clean, and inexpensive—perfect for busy kitchens in need of a little texture.
Related: Would You Put an Antique or Oriental Rug In Your Kitchen?
(Images: 1. Kirsten Ellis of Beaux Arts Photographie for Design*Sponge; 2. Leela Cyd Ross; 3. In My House; 4. Riikka Kantinkowski via Decor8; 5. Leela Cyd Ross)
Eileen's Beachside Oasis Kitchen
How to Make Fresh Pasta Dough in the Food Processor
Good Food to Share 
The Big Book of Small, Cool Spaces 
Apartment Therapy Presents 
The Eight-Step Home Cure 
Floral Drink Dispen...
Small Cool Lessons: Discovering Small Space Design Ideas That Really Work
Marisa and Peter's Playfully Eclectic Home
Rag rugs in the kitchen aren't new. My grandma used to have them.
The problem with them is that they tend to slip - a lot - on the tile floors in kitchens and can be a safety hazard.
I would suggest not using them unless you have some sort of non-slip backing on them.
I've got a couple of chefs mats (ugly) that are covered by rag rugs in the kitchen. They bring a nice cheerful punch of color and definitely make the room feel homey. Stains and spills don't show up as much and they're really easy to clean! Just throw them in the washing machine. LOVE THEM.
I have these all around the house, you can vary by colour and they can appear anything but shaggy. The only problem is that they can be very slippery. But that can be fixed with non-slip backing.
From when I was little until now, this has always been the only type of rug we've had in the kitchen. Get a non-slip backing and you're good to go!