I have no product name to give you but if you go to your local hardware or big box store and ask for stone cleaner that is all you need with a stiff brissled brush and a bucket. It isn't that messy. WE had a stone fireplace for years and about once a year (usually just after the chimney was cleaned and swept) we did a brush of the stone.

Thank you for the advice! I'll have to go get some stone cleaner and give it a try.