By http://www.ehow.co.uk/how_6556657_clean-amber-silver-jewelry.html
Cleaning Amber
Fill a small container with lukewarm water. Dip a portion of a soft flannel cloth into the water to wet it.
Wipe the amber gemstone with the moistened portion of the cloth.
Dry the stone with a dry section of the cloth.
Pour 1/4 tsp. of olive oil into a small container and dip a section of a clean, soft cloth into the oil.
Polish the amber with the olive oil. Buff with a dry portion of the cloth, then store your jewelry.