Well, it may work, but it isn't going to work /well/ It will full step only, which reduces the resolution of movement and can cause your motors to vibrate and miss steps at certain speeds. It also doesn't regulate the current flowing through the coil, so it isn't going to be as fast as it could be given a higher than motor rated voltage on the power supply and a regulating driver.
You can buy a pre-programmed PIC for the Linistepper driver for $6
which will give you microstepping, current regulation, and a high power half step mode which is really nice.
The linistepper design is open source, so you can make your own from the schematic or buy a pre-made PCB and stuff it from your junk box or scavenged parts. There are some board layouts available if you want to make a PCB yourself.
If you are going to buy the parts, you can't beat the kit price of $30 each (quantity 3)
http://www.ecomorder.com/techref/eco...p?p=416015&q=3 it will cost more to buy the parts at a distributor than to just buy the full kit.
Of course, that schematic you have will cost less in parts, and you can probably build it on a whiteboard. If you don't fry your TIP122's you can re-use them in a Linistepper when you aren't happy with the performance from this one. Don't connect or disconnect the motor with the power on!
Good luck anyway and let us know how it comes out?