I've had success with baking soda. Sprinkle it on, add water to make a paste. Scrub it in a little then let it sit. I think I waited at least 15-30 minutes. Scrub and grinse (I used a rag to wipe the paste up and rinse). It's messy but worked really well.

There are also special tile and grout cleaners made by the tile industry that are acid based. That is hard on the grout so I'd try something milder first. Vinegar is a mild acid and is often used as a first resort before going to the heavier duty stuff. I believe I also recently saw a non-acid grout renewal cleaning product made by one of the tile product companies. I would stick with their products and not try supermarket stuff.

I might even be tempted to pour some vinegar on the baking soda paste before scrubbing and let it bubble up. That mixture is used for cleaning some things. Googling, I see that people have used it on grout and say it works well.

Here is a link that might be useful: vinegar & baking soda

You can put warm water in a spray bottle with a little vinegar. If you use the water and vinegar mixture, make sure you dry the floor as you go.