The notion that mouthwash makes an effective cleaning solution for smelly stainless steel coffee cups is complete nonsense. Don't believe this, even if you hear it from a very practical sounding lesbian who watches Martha Stewart. In fact mouthwash is practically toxic as far as your cup is concerned.
Rather, try the better advice given by a former barista (barrista??) who is now a mailorder marketing specialist:
use baking soda and scrub hard.
5/5/01: saturday smelly cup update: the smelly cup has been through the dishwasher three times; soaked in baking soda for four days; scrubbed; and scrubbed again.... and after nine days, still fucking smells like Listerine.