The answer is : it depends.
The real thing to worry about is drying the board out properly and not letting the dishwasher get too hot, which can warp the plastic. If you do decide to run it through the machine, you should dry it out with a blow dryer and leave it upside down on a towel for several days before trying to use it.
But the dishwasher is overkill and for cheap rubber dome boards, can spell the end of the board for a variety of reasons. One thing I've found is that the board will be washed clean of lubrication and the keys may start sticking of have poor travel. Instead, I recommend removing the keys individually and tossing them into a ziplock back with a tiny bit of laundry detergent and water. That will take care of all the grime on the keys. Then clean the inside of the board with a slightly damp cloth and wipe carefully, using q-tips or something for hard to reach places.
One of the advantages to a buckling spring board, such as those made by Unicomp, is that they keys are designed to be removed with ease and cleaning the board by hand is a snap.
And yes, I will plug Unicomp in every keyboard thread. It's my duty.
