Wool is washable, you just can't throw it in the washer and dryer like a towel or a pair of jeans. Depending on how large and how thick your mat is, you can put it in a container of room temperature water with a little bit of shampoo dissolved in it and gently work the water through the wool. You can use a sink, a dishpan, or the bathtub, depending on how big your mat is. After you are satisfied with the level of cleanliness you have achieved, drain the water, remove the mat, and refill with room temp clear water. You can use a vinegar rinse if you wish, however, end with a clear water rinse.
The amount of agitation you use should be as minimal as possible, and to dry it you should (again depending on size and thickness) roll it in thick towels, or sandwich it between layers of towels and apply pressure to remove as much moisture as possible. It is not advisable to put it in a dryer, although if yours has a sweater holder, you might want to try that. (Wool that has long fibers and has been well felted will probably live through about any wash/dry routine, but many of the wool pads I've read about seem to be made of cut fibers, and I wouldn't chance it. I have this image of globs of wool sticking to the washer and the dryer filter...)
I try to only wash woolens in the summertime if at all possible, so that I can put them out in the heat in the shade after the towels. I live in a hot, dry climate, so its like a dryer and works well. If you heat your home to a toasty level in the winter, your mat will probably dry just fine if you lay it on a dry towel, after the rolling or pressing step.
In a former lifetime I sold a lot of woolen items, and have a lot of experience washing and drying them.