There are cube kits like that that use WS2812 type LEDs that would work.
I love cubes here are 2 that are easy to make with Pixelblaze:
- Walled cube using 8x8 panels. 3D printed corner clips hold the panels together. 8x8 WS2812 LED Cube
- Volumetric cube using LED pixel string. Many variations of this are possible. Pixelblaze Volumetric Cube