If I may suggest an alternative. Tiffanys sells / provides a complimentary solution for cleaning their fine jewelry / diamonds when you purchase them. That solution is actually very effective. I usually soak my diamond rings in the solution for a couple hours (I have realized soaking it overnight makes it shinier but not sure if this is advisable, but I do it pretty often and don't see anything happening to my diamond or band). When you're done, use the brush provided to scrub it AND you can use the Tiffany silver cloth to polish too. I know they use the ultrasonic machine with their solution and the cloth back there.
Other option is soap and water and use a brush to scrub as a daily routine. I sometimes find that soap and water whilst washing my hands always cleans it up and does bring out the shine