Are they dumb? Isn't there a microprocessor in every keyboard?

Yes, and it is perfectly safe to wash a simple circuit board, as long as the rinse water is clean, and it gets dried reasonable quickly. Both conditions are met in a dishwasher.

Yes, sorry, I meant there's no battery.

Microprocessors (especially older ones) are _very_ hard to kill. They're a piece of silicon embedded in epoxy essentially, very inert. There are a lot of people who wash arcade PCBs in the dishwasher, not a practice I endorse, but it works.