Q:

Why do electronic circuits need their power supply to be DC, or rectified AC? 

I'm pretty sure the answer to this should be obvious - even to me ...
It's been a while I've been reading and attempting to construct simple electronic circuits.  The supply almost always is either through a battery/cell, or AC duly rectified to DC.  
I have the impression electronic circuits rarely ever work off an AC supply.  

Is this impression correct? 
If yes, why is an electronic circuit rarely (if ever) powered/biased using straight AC?  

A:

Your impression is correct. 
AC flips polarity very often.  Most semiconductor devices have been designed with very particular polarity conditions in mind.  If one were to reverse the polarity between two inputs on many different components, you would get magic smoke coming out of it. 
For example take mosfet operation.  An NMOS is expected to block current from drain to source until the gate is activated.  If you have AC across drain to source, then it allows current to flow backwards (losing gate control). 
It's not impossible to design an electronic circuit with active devices around AC, but it makes it much more challenging.  I should also qualify this statement: many electronics circuits do have signals with AC components to them, but they usually have a DC offset to them to prevent the polarity switching.  That would be something like 5VppAC with a 2.5V DC offset would be an AC signal from 0-5 V. That remains the same polarity at all times even though it's passing an AC signal. 

A:

A keyword - in context with the question - is LINEARITY. 
Let us take the simple example of an electronic circuit, which is widely used: Transistor-based "linear" amplifiers (lumped circuits or integrated).  
Transistors are highly non-linear devices.  
In order to produce sinusoidal output signals exhibiting low signal distortion it is necessary to use only a certain part of the input-output charcteristic.  Hence, we use a dc voltage/dc current to BIAS the device at a suitable operating point (within a limited quasi-linear part of the input-output function).  Then, the input signal causes a swing around this operating point resulting in an acceptable output quality. 