I am making a very similar circuit. Attaching 3 LEDs in parallel and uses 9 columns of these to make a 3x3x3 cube. Problem is I cannot attach individual resistors to each LED. Hence, is there a solution whereby I can use 1 resistor for every 3 LEDs in parallel, use a transistor to supply approx 60mA of current?
You need 70 suitable resistors (one for each LED) and a transistor that can take the current. Assuming you have to work with 3.4V, you will need a transistor with a very low internal resistance, preferably a MOSFET with just a couple of mOhm. If you have 5V available, just use a simple 2N2222 with about 270Ohm base resistor.