I got a great way to clean grout from my cousin.Just mix warm water and some baking soda, making a paste.I use a small soft bristle brush and apply paste to the grout lines. Let it sit for about 30 seconds and wipe off with a wet cloth. It takes time, but your grout will look like new.
The easiest way I have found is to rent a steam cleaner. You don't need any chemicals and it is much easier on your back and knees! Simply steam the floor and then mop up any excess water.