This is the advice from a specialist Marcasite Jewellery Maker:
Marcasite Jewelry is also easy to care for. Simply avoid contact with hair spray and perfume, do not use any liquid polish as it may damage the jewelry, never use a vibrating cleaning machine or a sonic cleaner. Gently buff jewelry with a clean polishing cloth or a soft, white cotton fabric only. That's all there is to it.
Then I found this elsewhere:
Marcasite Jewellery should only be cleaned with a mild soap and water solution, and then wiped gently with a soft cloth (preferably cotton). To keep the naturally dark areas, be sure to clean it gently. This type of cleaning solution will be fine on whatever stones the jewellery has set within it.
And then this, that I think is probably the best way forward:
Care of marcasite jewellery
Marcasite is often secured in a setting with jeweller's cement. This feature requires special cleaning.
How to clean:
Wipe gently with a damp cloth that is soft and clean.
Dry with a soft, clean, absorbent cloth.
Air dry on a towel to ensure complete dryness.
Do not use hot water, a steamer or an ultrasonic cleaner.
Do not expose to cleaning agents or chemicals.
Do not expose to extreme heat.
Do not soak in water