Karen: I'm a new yogi. Very new. I began taking an instructional beginner's class in Ashtanga Yoga that meets once a week for eight weeks since last June. Our teacher actually brought up the issue of sore wrists with our class with regard to the downward dog position.
First thing she reminded us was that the fingers of the hands are spread out on the mat when we are down. The weight should be balanced mostly within the fingers, and the upper pads of the hands (the place where we generally develop callouses if we work with our hands). Often times people will roll the weight down into their wrists instead of using the whole hand for a solid foundation. Make certain that you are using the whole hand. Think of them as suction cups on the mat.
The other thing to keep in mind is that you must keep your shoulders pulled back and squared and pushed down from your ears. (If it's the downward dog position that is bothering your wrists) This minor adjustment can also help to alleviate the pressure in your wrists.
If you have the opportunity to take an instructional class you may find it very helpful just to get your positioning correct. You could even hire an instructor for just a session. It may help so much as you continue your own study with the aide of videos, books, and websites. I'll also share with you one of my current favorite books if you are interested in Ashtanga yoga, "Ashtanga Yoga, The Practice Manual" by David Swenson.
Good luck.