I'm assuming that you're referring to a mechanical dishwasher and not a member of your household. In either case, you can put an acrylic stem in your dishwasher, but in the latter case it may be a bit uncomfortable, embarrassing, kinky, or illegal. As long as your mechanical dishwasher is equipped with a door, you should be able to place your acrylic stems in it.
Personally, I just run warm water through my acrylic stems and scrub them out with a couple of pipe cleaners that have been dipped in a scent free dish washing soap.