PaulS:
or that 5V is important for some reason.
You are trying to imply that it isn't?
Given Jeenodes are running at 3.3v and regular Arduinos at 5V I had the impression that the voltage could be anywhere in the range from 3.3v to 5v and from the thread I linked to, perhaps even a wider range than this. Perhaps you're alluding to something bad which happens when the supply voltage changes while a sketch is running. Perhaps you're trying to underline my ignorance. I'm a bit confused by your question.
PaulS:
I don't think the exact voltage matters for us, so long as the microcontroller is running stable.
There is a big difference between exact voltage and consistent voltage. Batteries will give you neither.
It may not be clear from my wording, but these are unknowns for me - my main reason to raise it on the forum. If you know a reason why regulating the power supply to 5V IS important, please share. Equally if you have notions of 'exact' and 'consistent' for voltage which corresponds with the well-functioning of ATMEGA328 projects when powered by batteries, then I'm all ears. The insights promised by your post could mean I have no choice but to include a regulator. However, I don't think I can guess what those insights are, yet.