Actually you can use neutral detergents (like car wash soap or dishwashing liquid). The important thing is to dilute them 1 capful per gallon and rinse over with clean water.
I dont know what color your grout was before, I'm asumming white. More than likely your grout wasn't sealed before and the water acted as a carrier pulling the grime into the dry grout like a sponge or your soap contained phosphoras which breeds mildew (it's actually used as a fertilizer). This is why it's generally not recommended to use soap to clean tile.
If you have'nt tried bleach I would give it a shot. Do a small area in under the refrigerator or stove and see if it will work. Make sure you rinse it over after cleaning it. Mix 1 cup of bleach per gallon of water. Let it soak in for a few minutes. Put a fan on it and let it dry comepletely.
If it works, look into sealing the grout. Try this link for an easy to apply sealer.