When I remembered using the Steven James dryer agitation method for metal etching, I decided to try something similar to clean my diamond wedding rings as well as a diamond tennis bracelet I seldom remove. It worked like a charm!
When you are ready to do laundry, find a short wide mouth jar or small plastic container with a lid to use as a cleaning container. Place jewelry in the bottom, cover with an inch of warm--never hot--water, and then put a drop or two of Dawn dish liquid into the water. Replace the lid. There is no need to agitate the solution. The dryer will do this for you. Place the jar on the dryer. When the dryer is on, it provides gentle agitation and a bit of warmth. Cleaning can be sped up with the addition of gentle heat. I clean my jewelry when I am working on a metal clay piece and am using a dehydrator. A mug warmer would also work. If you feel the need, do a quick swirl of the jewelry in the water between loads of laundry. When done with laundry, the jewelry is CLEAN! A quick rinse and the pieces are ready to wear again. Note: If rings are especially clogged, they may need several laundry days of cleaning to finally remove all that clogs the settings.
Do not use this method on any gemstones that must be treated with great care, such as pearls, mother of pearl, coral, and turquoise. The length of time in a soapy solution could harm the stones. Of course, cleaning with soap and water for a short time and then rinsing and drying is just fine for these stones.