So definitely no boiling water or dishwasher. I don't clean my mouthpieces in anything warmer than tepid water.
If it's just basic cleaning, water, liquid soap, and a mouthpiece brush will do it. But if crud and calcium deposits are built up on it, try this:
Take a drinking glass. Create a mixture of half water and half white vinegar. Put mouthpiece in for 20 minutes. If it's a clarinet mouthpiece, don't submerge the cork. The diluted vinegar is a mild acid that will eat through a lot of the crud. Run mouthpiece under tepid water and use a mouthpiece brush to scrub the inside. If deposits still remain, return to the vinegar mix for 5-10 minutes and try again.
This may initially discolor the mouthpiece to green or brown. Rubbing a little olive oil on it should bring back the color.