My method is just to keep in mind all the overall shapes in a composition. Most people don't overlap things enough. Here's a great post by James Gurney on overlapping: [link]
But basically I keep in mind all the shapes--whether it's a girl shape or a lion shape or a teacup shape. It helps to have one big shape, a medium shape, and a small shape, and to avoid asterixes... points of tension where two shapes don't overlap quite enough (creating an "asterix" in the drawing stage). Not exactly a rule of thirds, but three elements tend to do better than two, which can be a bit imbalanced. Or if there are far more than three elements, then grouping them into three groups of elements (such as in a battle scene) can be useful.
Anyway, these are the things that are most useful for me to create pictures... hopefully there's a nugget or two in there that's useful to you.
Of all the aspects of art, you ask about the one that's the hardest to explain, even moreso on an internet forum.
I don't use the rule of thirds in my work, it fits more into photography than drawing.
What i do is make little sketches, amphasizing colors over shapes. I fill areas with colors so i have a basic idea where the "darks" and "brights" should be, again, no shapes. Then i make a bigger sketch, again building it all up by coloring- the shapes appear on their own in time. Making contrasts of dark and bright colors can do well for composition, althought it's not the ultimate rule. Try doing a research of your own and use the tutorials others have put here.
Hard question indeed since I'm still learning this as well lol. But I'm sure you've studied all those other ways like form, grids and rule of thirds etc. But I think there is a better way of going about it. I'm in no way of pioneering this but it help drastically understand the concept. Study from people who do this for a living! Study from your favorite films. The people who film don't just happen to film in some random area, they have a plan and have done composition thumbnails to get good shots. So watch some of your fav films and pause the film and do thumbnail sketches of the major portions that make composition, aka background, middle ground and foreground. If you notice all shots have some sort of this relationship. Sketch the scene, and try to understand why the director chose those shots. Trust me it helps a bunch, the more you do them. Hope I helped.
Make little quick sketches with different compositions. Look at how other artists work with compposition, learn the rules, try to break the rules aswell. you can do centered pieces, but you can try to go away from it aswell.
I prefer the grid system. Also, when I'm doing thumbnails and trying to figure out which one I want to pursue, I turn the paper upside down and sideways to see if it's still interesting. Diagonals are also great for creating dynamic compositions, but I'm not saying make every piece skewed and diagonal.