Perhaps try using props to alleviate pressure? Also, you could modify down dog to Dolphin by using your forearms instead of palms.
I'd suggest trying a different type of yoga practice for a little while and see if that helps - maybe Ashtanga is too intense for you right now? At least it would rule something out if you try a more gentle, restorative practice and have no wrist pain.

There are exercises you can do to increase wrist strength outside of yoga, so maybe check that out too, do some googling.
FWIW, I broke my wrist back in high school playing basketball and always had problems after that - couldn't do push ups, and sometimes even opening a door the wrong way was very painful (flat palm on the door). I started climbing, and using those grip strength tools (the rings or the squeezie bar things) and it seemed to help a lot. I am now able to do pushups and have very little pain in my wrists.

Barring any of that, maybe see another doctor - a sport-specific one?

I was having intense wrist pain as well, both wrists, off and on for many years, but mostly on for about 6 months. I started taking private yoga lessons, and through subtle adjustments I am now pain free, and doing more that I ever thought possible. I don't do ashtanga yoga, but a typical yoga practice for me will include several sun salutations. I'll try to explain this as best I can - hopefully it will make sense. When in downward dog, check your hands to make sure your index finger is pointing straight ahead and your fingers are spread. but your thumb should be close to your index finger (not touching, just closer rather than spread far apart). Move into downward dog, and make sure your hands are flat (sometimes there is a tendency to "lean" on the outside of the hand). Really press them flat and create a sensation of "moving" your hands forward. Of course your hands don't really move - it's just a sensation of moving them forward, kind of helps to lengthen from your fingertips to your shoulders and take the pressure off the wrists. As far as chaturanga goes, check to make sure you are doing this correctly - I'll leave it to youtube or yogajournal to explain that - I just know that I was also doing that incorrectly but not sure if that was contributing to my wrist pain. Also, the other person who responded suggested talking to your instructor, I agree that it a good idea. Good luck, and hope this all makes sense!

I am actually going through the same thing right now. I think I actually strained it attempting titibasana/firefly. My short arms had trouble with that one. Since then I have pain going from up-dog to down dog. I have been trying to adjust my hands so that I am really putting the weight in the thumb and forefinger area as it should be. it seems to be helping but I think I might pick up a small wrist wrap just for a little more support. Stopping regular practice just isn't going to happen.

Also before class i have been stretching my wrists out a lot and it seems to help. My favorite wrist stretch is to go into tabletop but then turn the hands around so I am palm down with fingers facing me and slowly lower my butt to my heels. Then I just sit in the stretch for a few breaths. I think it is helping.

I've been doing yoga on a fairly regular basis for the last 10 years and I really, really love Ashtanga. Unfortunately, I've been having some extreme pain in my left wrist off and on for about a year now. I went to my doctor and she believes the pain is due to inflammation (they checked for carpal tunnel and some other problems, but its not any of that). Whenever I have these "flare-ups" I can't do a full or even partial Ashtanga session. Does anyone know of variations to poses like chatarunga, down dog, pretty much any pose that bears weight on your wrists? Are there adjustments that I can do while in these poses? Is it even possible to practice yoga with wrist problems? I'm not sure what to do exactly and I fear I may have to give up my beloved yoga practice. Please help!

I have tendinitis in my wrists so my wrist pain comes and goes. I teach and practice Vinyasa Yoga and once in a while I will have a really bad flare up when my schedule gets crazy. I found these gloves made for wrist issues for yoga /pilates. They have a gel wedge that takes alot of the weight off the wrist. They called WAGS (Wrist Assured Gloves) and they have 3 different levels of wrist support. http://www.wristassuredgloves.com/

I will say they are kind of annoying to wear especially during a rather sweaty Vinyasa session but when my wrists are really bothering me they def come in handy. (No pun intended lol)

*UPDATE*

Just wanted to give everyone an update on some things I have been trying:

1) I've been switching my normal Ashtanga routine with a totally seated Hatha one about once a week. I didn't think I would like it but its actually giving my wrists a break and challenging me with some new moves.

2) Found this clip on youtube for modifying ashtanga for wrist problems and it absolutely has helped from hurting my wrists even more while being able to do my fav yoga style. Please check this out if you love ashtanga and have wrist problems.

http://www.youtube.com/watch?v=SCbVKN9uiYo

Thanks for the update

You've already gone to the doctor and had things like carpal-tunnel ruled out, but talk to your instructor, too. There could be something going on with your alignment or technique.

I echo using a rolled up towel or some sort of wedge prop to take weight off your wrists. Looks like you are varying your routine and having success with that.

Also, keep fingers spread wide and really press fingers into the mat instead of pushing all of your weight through the base of the palm. The more you can push into the mat with your spread fingers, the more you will spread the weight through your hands, reducing strain on the wrist. I agree with others saying to avoid rolling out to the pinky edge of the palm. Concentrate on pressing through index finger and mound of inner palm. These are the things that have helped me build up my hand and wrist strength and stamina in arm balance type poses.