It is a LED cube, and your PCB surface is minimal to support control electronics - the cube itself should be, as the matter of fact, detached. The main problem detaching is obviously DC losses at low voltages, but my particular idea is to do try and detach, and tackle this new issue which is more sensible than one monolithic-rigid design around a base PCB. If your n-size cube requires only one common ground - then you just cut the required contact points in half, and by bending the LED cathode contacts you may further reduce the PCB surfaces. In practice you end up with two smaller but right-sized perforated boards.
My next logical step here would be to stabilize and balance the cube for rotation and 3D imagery - but this is not what you have asked for. You asked for one big perforated plain.