I started kicking around ideas for eliminating all the connectors completely. If I went back to the idea of stacked, horizontal LED boards, I could just put some via holes at each of the board and stack them using rigid wires. These wires would have to be hand soldered and the power and ground wires would have to be suitable sized to carry the worst case current load.

While this might look slick, the amount of hand assembly was onerous and I'd have to come up with some sort of jigs/spacers to keep the boards at the correct separation to make a cube. I think this is certainly achievable but I ultimately rejected it as being too complicated to assemble.