Rent a walk behind saw cutting machine, such as the one pictured to the left. These machines look far more intimidating than they actually are to use. You can rent this kind of saw cutting equipment at your home building center or at a contractor rental center. The larger the blade and engine, the easier the work will be for you. Rent the largest machine you can comfortably manuever in the space you are working. Make sure the blade is in good shape before you check out at the cash register.
Layout your saw cuts with a tape measure so that your trenches will be uniform in size. Mark the concrete with a pencil and then spray orange marking paint to outline the cut areas. The orange paint is easy to see and follow when walking behind the concrete cutting machine.
If you can hook your concrete cutting machine to a garden hose, you really want to do that. There will be a nozzle connection on the side of the machine. Without water, the amount of dust created is unbelievable. The dust is also so fine that it will inevitably get in your eyes, nose and mouth regardless of your protective apparatus. You will be able to alter the flow of water through use of a valve, which will help you to not totally flood your work area.
Line up your blade with your orange spray paint mark, turn on the brick machine and slowly lower the blade. There will be a hand crank on the handlebars, which allows you to raise and lower the blade. Crank it down slowly and let the blade gently sink into the concrete. You may not know the depth of the concrete you are cutting, so just keep lower the blade until the fender over the blade is parallel with the ground. Move forward a few feet and then turn off the machine. Use a thin piece of metal or the like to slide into the saw cut line and check to see if you cut all the way through. You'll know you're through the concrete if you feel yourself poking semi firm dirt, which will give as you push into it.
The concrete saw cutting machine naturally stays in a very straight line. So, once you are lined up and know how far to lower the blade to make a clean cut through the concrete, just march forward. Cutting is very slow and you have to maintain continuous pressure on the handle bars to help the blade chew forward through the concrete. Don't force it, don't push hard - this is a finesse game and you just want to help the machine do the work and not man handle it.
Once your saw cuts are complete, you will need a jack hammer to break up the concrete block machine for removal. It is possible to dice up the concrete and remove squares, but it is extremely hard to do unless you are a pro. It is also a lot more time behind the concrete machine, which really is not much fun. Jack hammering the concrete between the cuts is fairly easy. Rent an electric jack hammer.