There are three things to consider when choosing the value of your resistors, the LEDs, the 74HC574 that drive the LEDs, and the transistors used to switch the layers on and off.1)
If your LEDs came with a data sheet, there should be some ampere ratings in there. Usually, there are two ratings, one mA for continuous load, and mA for burst loads. The LEDs will be running at 1/8 duty cycle, so you can refer to the burst rating.2)
The 74HC574 also has some maximum ratings. If all the LEDs on one anode column are on, this chip will supply current 8/8 of the time. You have to keep within the specified maximum mA rating for the output pins. If you look in the data sheet, You will find this line: DC Output Source or Sink Current per Output Pin, IO: 25 mA. Also there is a VCC or GND current maximum rating of 50mA. In order not to exceed this, your LEDs can only run at 50/8 mA since the 74HC574 has 8 outputs. This gives you 6.25 mA to work with.3)
The transistors have to switch on and off 64 x the mA of your LEDs. If your LEDs draw 20mA each, that would mean that you have to switch on and off 1.28 Ampere.
The only transistors we had available had a maximum rating of 400mA.
We ended up using resistors of 100 ohms.
While you are waiting for your LED cube parts to arrive in the mail, you can build the guy in the picture below: http://www.instructables.com/id/Resistor-man/