Quantity   Component name
64 × 5mm LEDs Cheapies bought from amazon. Nothing really special here. I chose blue. https://www.amazon.com/gp/product/B00E6P8J1W
1 × Tinned Wire This takes solder well, is easier to cut than piano wire, and is rigid enough to hold the cube's shape. All of the wiring is the cube itself is made of this stuff. https://www.amazon.com/gp/product/B000IJXXV6
1 × ATMEGA328P-PU 8-bit ATmega AVR with I/O pins that can source and sink more than 20mA to drive the LEDs. Acquired for free as a sample from Microchip.
9 × 41Ω resistor Exactly was it sounds like. I'll probably use 1% tolerance in the end.