Here are two options to consider for your cube... Soldering Sunday LEDs have stout legs with a distinct pattern that might help prevent wiring errors.

Traces will have to stack vertically to prevent crossing connections... especially the signal.

Adafruit has some nice 'leggy' LEDs that can help simplify ground / power connection hook-up.

If the distance between LEDs is roughly the length of one of the legs, hook-up of power and ground is going to be easy peasey.

Stack each new level by matching the signal out of the lower level to the signal in of the level above. Add vertical connections for power and ground... completing the structure of the cube.

Be creative... and sketch out a couple wiring diagrams before you start soldering.