Use an old, softbristled toothbrush and toothpaste to gently scrub off grime. The toothpaste can be applied to a cloth to hand polish an item as well. If youre seeing scratches on your jewelry, its probably the toothbrush, not the toothpaste use as soft of brush as possible.
Alternatively, simply scrub with undiluted toothpaste. This, however, can be harder to rinse out of small crevices in your jewelry.