In general there are two different things to keep in mind, the proper wearing and the regular cleaning. And as this blogs says, proper wearing means not to wear your ring during housework or yard work, the regular cleaning of course depends on what kind of ring you have. On how to clean stainless steel I found the video shown below, however I can´t guarantee you that the shown way will work. Or you can use this stainless steel steam cleaner, which in my mind is a very expensive way and for this price you might want to buy a new ring. ;-)

How to Clean Stainless Steel Jewelry — powered by eHow.com