Mike,
What chemical do you use?
Might be worth looking at a product designed specifically for cleaning stone and tiled floors and in particular grout lines, something like Dynax Tile, Grout & Safety Floor Cleaner (£25 + VAT for 1 x 10 litre).
Alternatively the hard (blue) brushes for the Multiwash 340 might offer the solution. You need two and the list price is £83.69 + VAT each, although I'm happy to offer a 20% discount if you want them.
Simon
Industrial Floorcare Machines (UK)