You can clean a moissanite the same way you would diamond or any other fine gemstone. As with any fine jewelry item, you would not want to dump all of your jewelry into an ultrasonic cleaner at once to guard against possible chipping, but your moissanite jewel is perfectly safe under normal ultrasonic cleaning. (Unless you have pave set stones - we do not recommend using an ultrasonic cleaner on pave set stones, as it will rattle them loose over time.) You can also clean them with a commercial (non-acid based) jewelry cleaner or mild soap and water using a soft toothbrush.
CLEANING TIP: When washing your hands, allow the water to run over your rings to thoroughly rinse away any soap, this will help keep your sexy moissanite cleaner longer.