Before cleaning jewelry, consider its condition. Each piece should be inspected carefully for loose or broken metal settings. If you see any problems, it's best to bring the piece to a jeweler for re-setting and professional cleaning.
Note that opal and some other stones are easily damaged by impact or rapid temperature changes. Handle carefully. Avoid blow dryers. Opals, for instance, contain water and should not be allowed to dry out.
Emeralds and turquoise may be treated with resins or oils that can be removed by soaking them in detergents, leaving them with less luster. Use just a cloth to clean and buff.
Stranded pearls are not damaged by water, but the string itself may be. Dry it thoroughly.
Rhinestones and other glued stones may be loosened by immersion in water. If in doubt, use only a damp cloth.
If you use a commercial jewelry cleaning fluid, pay careful attention to label warnings. Commercial cleaner isn't appropriate for some stones and metals. Also, proceed cautiously if using an ultrasonic jewelry cleaner. Softer gemstones such as amber and pearl — and even emerald, a harder stone — can crack when exposed to ultrasonic vibrations.
Bleach, even in diluted form, should never be used on metals used in jewelry, particularly gold.
For information on a specific gemstone consult the Web site of the International Colored Gemstone Association.
References:
Jeff Bredenberg, editor, Clean It Fast, Clean It Right: The Ultimate Guide to Making Everything You Own Sparkle & Shine
Consumer Reports editors, How to Clean and Care for Practically Anything