Here is a video presentation of the project:
The dice are based around a Simblee module (that conveniently combines an nRF51 chip with a chip antenna). The LEDs are charlieplexed using an I2C GPIO controller (mostly because the Simblee module cannot source or sink enough current). There is, of course, an accelerometer to detect orientation, and a small EEPROM to store non-volatile data.
There is even a small magnetic switch such that I can eventually build a carrying case with small embedded magnets that will make sure the dice remain off during transport.
All the electronics are soldered onto a flexible pcb and then wrapped around the battery in a cube shape, so that the LEDs face outward. All the other components had to be on the other side of the board.
The case is 3D printed (Shapeways) and currently measures 17mm on its side (a standard die is 16mm).
It was very challenging, in a fun way, to design the pcb and the case at the same time. Because of the small space, I really had to constantly think how layout would affect the way everything fit together in space, and vice versa.