Confusion with ATmega328P and 3.3V/8MHz
If you are using an LED to indicate that your sketch is running, then that may be the problem. 

For example, if you have a fairly average red LED, it might have a forward voltage of 2.0v @ 20ma (refer to your spec sheet for your particular LED). Putting these values into the LED calculator, it would tell you that you need a 150 ohm resistor. You might have tested this, and seen the LED light up.

Then, if you drop the voltage to 3.3v, your LED might be either very dim, or (more likely!) not light up at all. Putting 3.3v into the same web page, it shows you need a smaller resistor, 68 ohm, for the same LED. 

P.S. - I just noticed your paragraph about the green LED - a "true green" LED might have a voltage drop of 3.3v, which means for 5v you would need a 100 ohm resistor (as described in your paragraph); for a 3.3v supply you would need a 1 ohm resistor instead. This is pretty certainly your problem