An easy and safe way to do that is to soak it in a mild solution of water and a few drops of dish soap once every couple weeks. After you remove your diamond from the water, use a new soft toothbrush to remove any remaining dirt (check the back of the diamond close to the metal- this is where the bristles come in handy). Reserve this toothbrush only for your jewelry cleaning purposes. And be sure to be gentle. Older settings and more intricate settings won't take vigorous scrubbing; be gentle so as to not damage your ring. If you're working over a sink, be sure to close the drain! Even better, rinse your ring over the bowl you used to soak and scrub it over. Rinse it gently and dry it with a soft cloth.
Do not use bleach or toothpaste and other household strong/abraisive cleaners.
While your diamond might be fine, they can damage the metal.
You should also consult a jewelry professional regularly (at least twice a year) to make sure that your stones are tight and that the metal is not getting too worn; this will help prevent you from loosing diamonds and/or colored gemstones.
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