UPDATE: This design has been superseded by my newest version, http://www.thingiverse.com/thing:213946, which is all printed and snaps together.
This started out as an exercise in using Greg Frost's awesome involute bevel gear script http://www.thingiverse.com/thing:3575, but turned into a rather pleasing worry-ball type thing. It's also clearly influenced by Greg's broken heart http://www.thingiverse.com/thing:4683, though this has octahedral gears rather than dodecahedral, so they all mesh with each other and spin. My favorite thing about it is that you can grab two opposite gears and twist them, making the center with the other six gears spin rapidly around.
Print the parts list. I used a stock TOM, printing raftless with zero fill (just a couple extra shells). Skeinforge 35 made some odd slicing decisions, but it didn't affect the outside of the parts. These are good parts to convince you your calibrated printer still needs some tweaking.
Use a screw to tap threads into the center block first, because it makes it easier to assemble. No glue needed, the screws should stay tightly in the center block while the gears revolve. Attach the gears, making sure the edges of the cube connect them. If the gears are too tight to spin easily, just unscrew them all a millimeter or two.
The OpenSCAD file is highly parameterized to allow easy tweaking (for different proportions, screw sizes, etc.). You can also change the ratio of teeth on the gears, but you'll need to put the desired parameters into gearopt.m and run in Matlab (or Octave) and copy the result into the OpenSCAD file. This is necessary because I couldn't solve the geometry with anything but an iteration loop, which OpenSCAD doesn't handle so well. I used the MCAD library for Greg's involute gears https://github.com/elmom/MCAD.