What is the best product to clean grout between tiles?

Try mixing 2 parts Bicarbonate of Soda mixed with 1 part vinegar (or lemon juice which smells nicer) into a paste and scrub in with an old nail or toothbrush. Leave for about 5-10 minutes then rinse off.

No nasty chemicals required. By the way - when you add the vinegar/lemon juice it will get fizzy. :)