The Rule of Thirds, Part 3--Points
Don't make the rule of thirds the sole aspect of design in your composition. In the above photograph of a Gulf Fritillary, there are many compositional concerns concerns to keep in mind. For one thing, I wanted a softly-focused but pleasantly arranged grouping of leaves in the background. For another, I wanted a couple white flowers in the frame. Notice in this picture, you could rotate the camera significantly while keeping the butterfly on the same intersection point. Because of the angle of view on the butterfly, many orientations would still look natural. Other than keeping the shadow side of the butterfly's body below the lit side, you have freedom to rotate the camera to frame the shot the way you want and still put the butterfly near the intersection of the rule of thirds. You should make sure take all that is important to you and include those in your composition, removing what is not important, and let the rule of thirds help you do that.
I sometimes hear people review photographs with statements like, "I like the way you followed the rule of thirds." Well, what makes the picture work or not work is not precisely that you followed or didn't follow the rule. The rule of thirds is a tool that can help you achieve your goals. Our critiques should actually be something like, "I like the way your use of the rule of thirds affected your composition in this way." This is probably what reviewers mean, but it brings up a point to keep in mind when photographing any subject. Use the rule of thirds to your advantage when it helps. It just so happens that it often does, but don't let your goal be to follow it--instead, use it to aid your compositions.