Hi L.,
Magic Erasers work great on grout, but if it's a big area, you'll go through a lot of them. Or, baking soda sprinkled on the grout and then spritzed with water to make a paste works really well. You can get a grout brush, basically a long, skinny stiff brush, which makes it easier. Beware that this is time consuming, and the baking soda takes a sweeping and several moppings to clean it off the tile completely. The plus side is that both of these ways are totally non-toxic and chemical-free for the little ones that are all over the floor all the time.
By the way, I'm a speech therapist, too. Have you considered starting a business and seeing kids in your home so you can be home with your little ones? It's something I've considered...