Hi MatLepine, welcome to PF!
No, it wouldn't work. For the contraption to work, you'd need some elements to rotate at a different angular velocity than others. That is to say, the outer weights would have to complete the rotation around the central shaft either faster or slower than the shaft itself.
All parts of Eath have the same angular velocity(360°/day) w/r to the axis or rotation, though.
The somewhat counterituitive physics to understand here is the Newton's first law of motion for rotation, which states that in the absence of external torque(i.e., forces changing rotation) the rotating body will either not rotate, or rotate at a constant angular velocity, forever.
In everyday life we're used to thinking that rotation requires a force to maintain. Windmills need wind to keep working, ice skaters eventually slow down in their piruettes, etc. In all those cases there's a force(friction) acting against rotation.
With Earth, there is no such force, at least not on the scale useful for energy extraction(there are tidal forces, but that's another kettle of fish).
What one could try to do, I suppose, is use the Coriolis force(
http://en.wikipedia.org/wiki/Coriolis_effect#Applied_to_Earth), for example by swinging a giant pendulum in a vacuum chamber, and somehow extracting the energy as it gets deflected.