I perused around on the internet and found that baking soda with a dab of warm water to make a paste proves the perfect concoction to clean your engagement ring. I tried it, and oh la la, that thing is just like new. I mean, it kind of still is new, but this helps it maintain the newness in my eyes. You just rub the paste all over the ring with your fingers, no towel needed. Then you just rinse the ring off and dry it on a towel. Economical and easy, give it a try!
P.S. Works well on other jewelry, too. I've already tried it to be sure!