Some places will want specific shoes, but in my experience most don't care and if they do care it's only that your shoes are slip resistant. Slip resistant is a really, really good idea whether it is required or not.
I'd higly recommend restaurant specific shoes. They are designed to be easy to clean, comfortable to stand in for long periods, slip resistant, and easy to remove quickly. I can't stress how important the last two are. I've personally dumped a hotel pan full of boiling water into my shoe, and if I wasn't wearing kitchen clogs that I could pull off immediately, I probably wouldn't be working in a kitchen anymore. Slip resistance has saved me a few times as well. Cleanability will also be a big issue. Scrubbing the menagerie of different kitchen gunks out of the little crackes and crevices in most shoes is an exercise in futility.
I know a lot of people that like Crocs. I haven't used them so I can't speak for them. Wal-mart has Tredsafe clogs that are as cheap as it gets and will work, but are uncomfortable without insoles and wear out fast. My favorites are Klogs brand. They're the most comfortable that I've used and wear really slowly, making them good bang for the buck. They'll set you back about $50 or so and can be found at uniform stores and restaurant supply stores.