I would say it has more to do with position than phase. Just like the tides, the moon's position relative to the observer is what it's all about. If the moon is overhead or underfoot at dawn or shortly after, I park it on a deer trail as they return from feeding. If the moon is right again during dusk or shortly before, I know I better be camped out near a feeding area. I don't even hunt in the middle of the day because I have to work, but I see plenty of action in the morning and evening if I follow the lunar position.
I personally think the phase thing is ridiculous. I'm a a computer programmer and an astronomy buff, so I know a thing or two. Think about it: what IS phase? It's the "age" of the moon on it's 29.5 day cycle. What the moon LOOKS like is unimportant. It's GRAVITY, however, is always present and is strongest when it's either over your head or under your feet. That's why it tugs the whole darn ocean in one direction as the Earth revolves.
I've been using this strategy for a few years, and a buddy recently told me about Hunt By Moon. It does all the calculations for you. Pretty cool.
I'm pretty much sold on by personal experience that the morning after a full moon is DEAD. Very few sightings the morning after a full moon. However the evening before the full moon night is a different story.