I read the blog linked to by BobYIL in the OP, as well as the blog linked to by gns, and the discussion in this thread, and have to say I've learned a lot. Artistic composition has never been my strong suite, having pretty much mastered the Rule of Thirds and not going past that. For the most part, though, I have always tended to just take pictures of things that interest me, and to heck with the composition theories. As a result, I have a few really good pictures, and a whole lot of not-so-good pictures. Luck of the draw, in other words.
Yesterday, based on what I have learned in this thread (which includes the idea that this is not all to be taken too seriously), I went out and changed the way that I compose. I changed positions to find a better contrast between a subject and background, and I skipped shots without the contrast. I looked for subjects falling on lines within the frame, and skipped some shots that had nothing on the diagonal lines. I did this just to see if it makes a difference.
At the same time, it made composing pictures more difficult and less difficult. It was more difficult because I'm breaking years of bad habits of taking pictures without real thought or understanding as to good composition. It was less difficult because there were some simple rules of geometry to follow.
It was enjoyable hunting for a scene with good composition, and hunting some more to see if there wasn't a better composition possibility nearby. I'll process the negatives today and see if it really made for better photographs.