One of my goals this year is to be able to do a real push up (NEVER have been able to, even when I was really light)!! So I have been doing the bent knee push ups almost daily, as well as planks and taking yoga classes that spend a lot of time in downward dog, so yeah, I've been feeling it in my wrists ... one of my instructors said something that I hadn't thought about before. She said that wrists need to build up endurance just like every other part of your body and there really isn't much "real life" activity for most of us that puts weight on the wrists. So assuming you don't have an injury or other reason that you really can't put weight on your wrists, it just takes time till the wrists are OK bearing weight.
One option that is offered in yoga is using closed fists so that the wrist is not bent. I've done that before, when my little wrists just give up, but that kind of hurts in a different way, on my fingers. Maybe mix the two postures until your wrists are stronger?
to you! I hope more people write in with advice about how to do good pushups...