Grab your tarnished piece and try it on in every possible combination cuz obviously you haven't worn it in a while. (Please choose only your cheapy-fakey pieces for this method.)
Then take it off your finger, dummy. It's easier to clean that way.
Grab the toothpaste you used to brush your teeth this morning (or at least I hope you used it this morning).
Plop some on your ring.
Smear it around and really rub it in. If you have an old toothbrush, that'd be handy, too.
Depending on whether you're cleaning one piece really quick or all your pieces at once, you may or may not end up with super cool mint hands.
Rinse off the toothpaste and check it out. This is after one rubbing. You might have to go through the steps a few times to get what you want.
This is after two scrubbings. Good enough for me!