Why is there no net current in a wire without a voltage applied?
Statistically, there are as many electrons moving in one direction as there are in the 180º opposite so there is effectively no net current.  What we know as "current" is the movement of more electrons in one direction than all the others (1D, 2D or 3D through a piece of metal).  That's how you can have "tons of free electrons" but no net currents flowing or measurable.   

The random agitation of those electrons has a name: thermal noise.  This agitation is proportional to temperature so you get more of it as you heat things up.  However, the average motion is always zero so you can never do any useful "work" or equivalently extract usable energy from the process.

This is in agreement with the laws of thermodynamics.