YOu can go to the Home Center and ask for Grout Cleaner. Trouble is, you don't know how DEEP the 'problem' goes. With that, most products are bleech-based. So be careful when you use it.
The only other remedy is to regrout. If you're handy enough, you can do it yourself. It involves removing some of the existing grout. ANd get a grout with a hardner/sealer built into it. You'll need to work faster w/it - just don't try to do too much at one time. Work a small area at a time. THere are TONS of web sites that talk about how to grout.
My best to ya and hope this helps.
PS: Here are 2 of just a few HUNDRED links that sell a grout cleaning product. Perhaps they're sold in the Home Centers. (You'll have to ask the Retailer and the Home Center.) BE sure to look for something heavy-duty because of the grease. Soap and mildew cleaner is like Tile-X. It's not for grease ...