Up to 70 LEDs on one digital output

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?