I move the camera around until the image in the viewfinder is in balance. I change lenses, zoom if possible, take half a step to the left, etc. Light, dark, shape, texture, color, etc. all have to compliment one another. This requires that I trust my 'eye' and also keep in mind the camera's dynamic range, color rendering, etc. (am I shooting E6, C41, or the Drebel?) I don't pretend to get it right all the time, but when I do I have a pretty picture that is a direct expression of my sense of aesthetic. Hell, maybe I could even call is 'art?!'
If the Rule of Thirds (ROT!) enters my head, I bash myself with a large rock until the thought leaves. Seriously, if photographic aesthetic could be boiled down to one or two simplistic geometric 'rules,' the whole exercise would rapidly become a boring waste of time. IMO, ROT simply replaces one lame-brain mode of behavior (unthinkingly Bull's Eying the subject) with another (unthinkingly placing the subject at a third intersection.) This is no way to be! I'm just waiting for the day some camera company introduces "Auto R.O.T.!" The camera already had multi-zone luminance and distance readings, it would be a simple thing to program the camera to prompt the user to move the camera until the closest bright subject falls on the ROT grid. The camera would then take automatically take the picture! Done, and ready to move on to the next photo cliche! "Introducing Canon's new flagship consumer model, the 600-D, now with Auto-ROT! Making photography easy!" I'm quite sure that a LARGE percentage of shooters would embrace and praise this "Artistic Breakthrough."
In short, it's not about "Rules and when to break them." (Knowing 'when' is just another rule, right?) It's about learning to 'see' what's around you, thinking about it, and fiddle-farting with the camera until you have translated that vision onto the image in the viewfinder. There are no easy 'rules' for doing so, nor should there be. It's supposed to be difficult, otherwise what the point and where is the reward? I think people are often afraid to trust themselves, and prefer the safety of ROT and other nonsense. If you don't get past this, you will never make pretty pictures that reflect your personal interpretation of beauty.