Get yourself 4 NPN, or 4 N-channel MOSFETS, and you can it this way, or use the shift registers to drive the anodes.
With all 16 LEDs in a layer turned on, could sink up to 320mA, be aware of that when selecting a transistor.
There are also limits on how much each Port (group of pins) can be sourced by the Atmega328, see the notes at the end of Table 29-1 in the datasheet, so that may impact your choice of resistor.
74HC595 is limited to 70mA total current, so another option is using PNP or P-channel MOSFETS as the current sources.
