The "kindest" way to clean jewellery is to make a paste of tumeric powder [ sold in supermarkets and all Asian shops] and warm water, not too thick, put the jewellery in and leave overnight and then rinse in warm water., either gently pat dry or leave on a tea towel to dry. no chemicals involved. Works wonders with gold & diamonds but havent tried with marcasite to be honest, test on a small bit first.
My pal swears by toothpaste applied with toothbrush and rinse but I have never tried it, a bit harsh for my liking.
Good luck.