Zircon is a naturally-occurring gemstone that should not be confused with cubic zirconia, a synthetic gem. Mined in many locations throughout southeast Asia, zircon is a relatively hard gemstone which can be chipped easily if mishandled.
Before cleaning zircon, 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.
If the settings are OK, the safest way to clean any sort of jewelry is to wash it in warm (not hot) water with a mild liquid detergent added. Use a bowl and work away from the sink to avoid losing jewelry down the drain. Soak each piece for several minutes (not hours) and brush gently with an old toothbrush to dislodge any stubborn dirt and oils. Then rinse with clean water and dry with a soft cloth or paper towel.
If you use a commercial jewelry cleaning fluid, pay careful attention to label warnings. Commercial cleaner isn't appropriate for some stones and metals. Ultrasonic jewelry cleaners may not be appropriate for zircon given the propensity of its facets to chip or crack when bumped against another hard surface.
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.