Cleaning limestone open fireplace. Best solution/product?

Are there any decent home remedies for cleaning a limestone hearth and surround?

We need to start doing ours as a thin film of grime has started forming, and I'd rather not have to use some expensive chemical. Would something like warm water and bicarbonate of soda do the trick?

The grime is also settling on the oak wooden furniture (TV stand, drawers etc.) so something that could do that as well would be a bonus.

Many thanks