You need a non-abrasive degreaser. Visit a Home Center and start looking around. You can ask almost any Brick Mason for info, or try to visit a Quarry or Stone & Gravel Yard for advice. Heck, there may even be a restaurant or 2 in your aread that has brick in the kitchen. For some reason ZEP is coming to mind. I'm not sure if is OK w/brick. If brick isn't mentioned on the container, you can still ctc the Mfgr. for feedback (either by phone, mail, e-mail, or whatever.) Just because brick isn't mentioned that doesn't mean you can't use it.
I'd use a nylon-bristled broom/brush. The key to this job is not to remove the outer 'shell' of the brick. Remember, it's a fired 'material, where the inside is 'softer' than the outside. If you remove the outer layer, you expose the softer inner layer. (And you don't have that much to work with.) You may need to do the floor 2 or 3 times. Just plan on at least 2. Once you've got it good and clean, then you need to do something about cleaning the morter-joints AND re-sealing them. Again, talk to the folks at the aforementioned 'locations'.
My best to ya and hope this helps.
Jay J -Moderator
PS: Brick is somewhat porous. Yes, it actually breathes which is why it's NOT recommended you seal it. (The joints are OK to seal.) Don't let the 'stuff' sit too long on the brick. In fact, when you're finished, I'd just clean the floor a time or 2 with plain water. DO dump the water as you go because you want CLEAN water throughout the job ...