There are a lot of posts about jewelry cleaning. Much of it is good information, but some can actually do more harm than good. Keep it simple. Avoid any harsh chemicals and do not use boiling hot water. All you need to clean your jewelry at home is a grease cutting dish washing liquid, a soft bristled toothbrush, a small bowl and hot water from the tap. Soak your jewelry for 5 to 15 minutes in the soapy hot water. When the water begins to cool, brush the debris out and check the progress. If the metal and stones are shiny, you're done. If there is still build up under the stones, use more warm soapy water and let the piece soak a little longer. If after several attempts the piece isn't clean enough, you may need to have a jeweler clean it.
They will use an ultra sonic to deep clean and a polishing machine to get the scratches out of the metal. At that point they'll also check your stones to make sure they are secure. Which you can do as well. With just a fingernail or toothpick, gently rub across the top of the stone, side to side, forward and backwards and see if the stone seesaws in it's setting. If it does, take it to a jeweler to tighten or tip the stone in place.