A couple things that I've been thinking about (or overanalyzing... LOL) -

Somebody mentioned that it's sometimes like riding a horse. Now, I've never ridden a horse, but I can imagine how it applies. On rough surfaces, or on things like the ramped transitions from the street to a sidewalk, I just go with it, not even consciously trying to point the bike straight. Left on its own, a bike will sort of keep control of itself (there's a vid out there of a racer losing control of his mountain bike, which kept rolling out of view from the camera). Get off the saddle, and let it buck forward & back like a horse jockey.

Also, as I think about it now, I ride with a pretty light grip on the bars pretty much all the time. The only times I grab them hard is when I'm standing and really pedaling hard, too. Otherwise, my fingers are pretty loose.