All you have to do is soak your cups for five minutes in white vinegar. Since it's acetic, it will dissolve the minerals in a snap. If stubborn spots still persist.....

gently rub them off with baking soda. Then rinse the glasses by hand and dry them with a lint-free towel before drinking out of them again.

If this doesn't work, chances are your glass is etched — which, sadly, means you're out of luck. "If the film is etching, which happens to some types of glassware in the dishwasher, that unfortunately is permanent and can't be removed," So you might want to wash your most prized vessels by hand in the future — just in case.