The rule of thirds should always be a STARTING point. Then go from there to get the shot looking good. Remember there are four points in the rule of thirds for the subject.
I'm just the opposite. I go to the rule of thirds when the thing I saw in my viewfinder doesn't seem to be as appealing once I get it on my computer and I feel the need to do some salvage work.
I do a lot of photo browsing. I sit with my breakfast and look through the top shots of the day at 500px.com, I browse thru Instagram profiles from profiles with shots liked by photographers I follow, or just plug something I'm interested in shooting into an Image search and just look. I try and latch onto aspects of the shots I like and stick them in the portion of my brain that I dig into when I'm out looking for something to photograph - when my eyes are
seeing and not just looking. I then frame in a way I like and I shoot.
Rules are made because there are natural ratios that tend to be pleasing to the eye. I actually prefer the Golden Ratio to the Rule of Thirds because it can be equally pleasing, and it doesn't look like you were shooting from a textbook the minute you see it. An interesting subject and attention to lines, leading and anchoring (i.e. get a solid vertical and/or horizontal), make for a solid composition 9 times out of 10 regardless of whether or not there's a "rule" supported by it because it's got a foundation to support the subject. When the photograph doesn't seem to work then chances are that it's lacking that foundation and you can sometimes salvage it by seeing if you can apply a composition rule to a crop in post that'll help you out. Composition should be second nature to a photographer.
See your photograph before raising your camera, frame it and shoot it. Yes, rules might be applied but it will be done in a way that is absolutely organic, not forced onto the scene - and there's a good chance that rules will be broken.
Before diving into photography I spent the better part of my artistic life as a musician. I studied composition and theory, chord structure, you name it, and while it made me a better "musician" it stifled me creatively. Yes, I could write songs, and more often than not "fix" songs that others I was playing with had written. But I marveled at some of my bandmates' work, because they would bring in these things that I never could have written because, well, you
just didn't do that in proper songwriting. Heck, one of them didn't even know what chords he was playing on the guitar and I'd have to figure it out to teach it to the piano player. Sometimes not knowing the rules and just simply knowing and doing what you like is head and shoulders above what the textbooks will teach you. If you struggle figuring it out on your own then learn the rules but don't become a slave to them.