A simple solution of ammonia and warm water should be enough to get your diamond ring sparkling like new. Use a soft tooth brush and gently brush away any accumulated gunk.
Note: This is NOT a good idea if your diamond has been fracture-filled. At Mervis, we don't sell any fracture-filled diamonds, so if you've got a Mervis diamond, the ammonia is safe. Otherwise, a mild liquid detergent should suffice.