This poses a real challenge for a lot of people. What to clean? How? How often? There are so many choices out there, from rouge cloths to ultra-sonic machines that it can be overwhelming. Here’s what works for me:
A final thought -- I highly recommend professional cleaning for pronged setting, very high end pieces, and for heirloom or vintage pieces. It's far better to be safe than sorry.
For a link to the article on Hardness of Stones -- Moh's scale -- click here.