The way I wired the MC with the shift registers looks pretty much like this : http://picprojects.org.uk/projects/lc/cube555cS.pdf except that each layer has the cathodes in common and the MC is an ATMEGA1280. From what I understand in the program in my original post, the three pins for each layer flash a LOW signal to light the LEDs while the shift registers send out the HIGH signals.
Grumpy_Mike, what do you mean when you say the chips sink the current? In the datasheet, it says that each output can source 5 to 100mA (STP08CP05 Datasheet(PDF) - STMicroelectronics).