Hmm, so let's take an example. ATMEGA328P-PU running on 3 AA's and no regulator. Is there something in the data sheet which advises against this?
An AA battery gives you 1.5V so normally you have 4.5V but when they are very fresh you might get 1.8V from them so that will be 5.4V.
Now look in the data sheet at section 28, and see the absolute maximum operating voltage. This is the voltage you must not exceed. It says 6V so you are OK. Now a battery running down might go to 1.2V so this would give you 3.6V to work from.
Section 28.2 shows you the recommended operating range 1.8V to 5.5V so again from that point of view you fine.
However, section 28.3 shows you that the maximum speed of the clock is related to supply voltage, so you then have to plug in your systems clock frequency and see if it will run at the range of voltages you want. It looks like this is the limiting factor for your design decision of using three AA batteries.