This is another one of those solid, yet highly porous, materials that holds grime well. Exterior brick can be pressure-washed and rinsed, but that’s not a practical solution for exposed, untreated interior brick.
First, remove loose dirt by vacuuming the surface well. Clean the brick using a chemically treated dry sponge unless the surface is especially rough.
Deep cleaning will require water, and that means protecting adjacent walls and nearby floors and furniture with drop cloths or plastic sheeting (be careful on the floor – plastic is very slippery when wet). Wear gloves and eye protection, then gently scrub an alkaline solution or degreaser into the brick. Allow it to soak in and work for several minutes to loosen dirt deep within the porous surface. Blot with an old terry towel or other absorbent cloth. Scrub the surface again in two directions at right angles to one another, blot, scrub with a rinsed brush, and blot dry.
Once clean, consider painting the brick or applying a masonry sealer. Either will make future cleaning much easier. Clean painted brick with the same methods used on any other painted surface.
Reference:
Don Aslett in The Cleaning Encyclopedia: Your A to Z Illustrated Guide to Cleaning Like the Pros.