Below is the Sonobe unit, designed by Mitsonobu Sonobe, which I learned from Kazuyo Inoue. Above is the capped icosahedron which she made for me from this unit; you need to make about 30 units for this, but you only need six if you want to make a cube.
Since this must be very well known, I'm only putting a very brief picture of how to make it. The second line of instructions gives a little more detail for the last step.
Just in case you want more details than above, you can find some more diagams here and here. For the model pictured above, you need to make 30 units. Start off by putting groups of three together to make the "bumps"; then work out how to put 5 "bumps" together, and so on.
Kazuyo first showed me a kind of stellated icosohedron made from this unit; and there are lots more things to be made from it. I've been playing with working out what kinds of nice ways are there to arrange coloured peices to make the stellated icosphedron; eg, if I take six different colours of paper, what kinds of nice ways can I use them to get patterns with lots of symmetry?