I am *so* not an expert, but I always use either dumbbells or push-up handlebars when doing push-ups. Doing them flat bothers my wrists, and I found it easier to make adjustments than to try to fix it.

That said, I do agree with everyone else that if you're having pain in your wrists at other time, getting it checked out is the thing to do. If it's something potentially serious, it'll be a lot easier to fix it if you catch it early.