It's surprisingly simple -- toothpaste!
That's right. Just put a tiny amount (pencil-eraser sized blob) of your toothpaste on your fingertips and rub it all over the jewelry. You only need to rub it in for about 20 seconds. You may also want to use a cotton pad or something if it's particularly tarnished. Then just rinse off and you'll see an insane difference!!
Before and after side-by-side shot of a pair of my earrings: