What I do:
I place my pieces (about 4 small pieces max, but 1 at a time is better) at the bottom of a small glass, I then pour some dish soap on top of the pieces enough to get in the grooves that contain the dirt. I then let the water run at full strength until it becomes super hot, and then place the glass with the pieces directly under the streaming water (you want to make sure their is good water pressure). Fill the water half way in the glass, swoosh the pieces around a couple times, and let the pieces sit in the hot water for about 10-20 mins. Once the time has passed, gently rinse them with clean water, and dry with a small had towel, voila! Beautifully clean jewelry!
TIP: You may not want to do this with fake pieces that get destroyed with water, I only use this trick with my real jewelry. You also want to make sure you have a small cage strainer in your sink in case you drop them by mistake in your sink! If you haven't cleaned your jewelry for a long time, you can also use a small toothbrush to rub off the gunk after pouring the dish soap on it, and before letting sit in the hot water.