Can I produce radio waves by waving my hand? I learned that EM waves are caused by the movement of charges (e.g. electrons), because they have an electric field and the change in the particle's position doesn't update the field instantly all over space but propagates at the speed of light.
With the same reasoning I also learned that thermal radiation is caused by the random movement of particles in any object with temperature > 0K.
So if I wave my hand at 3hz (with all the charged particles in it) am I producing an ELF radio wave? Is it that the energy of this wave (low frequency, low amplitude) is just to low to produce any effect? Or is there something wrong in what I've learned?
 A: In short, yes, but very weakly.
Long answer:
Ordinary matter (like your hands) is composed by atoms, molecules, or ions aggregates (e.g., table salt). Since the lowest energy state of these systems is charge-neutral, it is energetically favorable that any piece of ordinary matter reaches a charge-neutral state. However, your hands can still have a very small, spurious charge imbalance. As a consequence of that, waving your hands can produce very, very weak EM radiation (and perhaps gravitational waves, as Javier has pointed out).
Just notice that the contributions to the EM radiation of the huge number (multiples of the Avogadro number) of electron and protons in your hand cancel each other over a very short distance, and in practice a net EM radiation is produced only if the net electric charge of your hands is non zero. 
Of course you can increase the magnitude of the EM radiation by increasing the electric charge of your hands (don't try this at home).  
This cancellation does not occur in the case of gravitational waves. 
By the way, I think that for this reason, the gravitational waves produced by your hands are stronger than the EM radiation in natural units, that is, if the physical units of measurement are set such that the gravitational and electromagnetic coupling constants are $G=\alpha=1$.
A: No.
Firstly, as it has been noted in the answer by @syntetico, the net electric charge of hands is zero, i.e., moving one's hands creates no electric current, and consequently no source term in the Maxwell equations.
Secondly, even if one holds a charged object in hands (or somehow caused charge imbalance on one's hands), these charge oscillations will not result in propagating waves, due to the impedance mismatch with the environment where these waves have to propagate. The rule of thumb for creating propagating waves is using antenna that is about half-wavelength long. For 3Hz waves one's hadns are too short, whereas for having wavelength comparable to the extent of one's hands one needs to wave them with a Megahertz frequency.
