I would look into the masonry anchors available to you. The last thing you want is them falling. Give consideration to how much weight there is. Someone would have to confince me that the brick will not crumble.

Use anchors into the mortar joints (not the brick itself).
For heavily loaded cabinets steel expanding anchors should be used.

It is usually much easier to attach a wood ledger to the brick, then attach the cabinets to the ledger.
With a wood ledger you can spot fasteners as required without worrying about the cabinets.

Tapcon screws work great. This is how I fastened my cabinets to concrete walls in the basement. Drill a hole in concrete and these screws will screw right into the hole. Makes it all much easier.

Brickeyee, this is not a disagreement, its curiosity, why is the mortar a stronger hold then the brick it self.

Tapcons are great in concrete, but i wouldn't trust them in block. Block is much more pourous and not nearly as compressed as concrete. I would use sleeve or wedge anchors.

There is usually no load rating for the bricks and they are much more subject to cracking from drilling and using expansion anchors.

The same applies to Tapcons.
They are designed for concrete, but can be made to limp in mortar.
Using them in brick or black is a crap shoot at best.

I realize this is a very old thread and perhaps I should start a new one, but I've had this same challenge. I mounted my kitchen cabinets into 100+ yr old brick using a masonry bit, and wall anchors. I also used canned air to blow out all the dust from the hole and then dampened the inside of the hole, then inserted Gorilla glue and then tapped in the anchor and attached the cabinets. I've had 2 cabinets fail and begin pulling out of the wall (one quite suddenly and dangerously and am now truly perplexed as to how to mount them to safely stay put. Any additional advice would be greatly appreciated.