I was going to say what Liosama said, yeah. You can almost always tell an onyomi from a kunyomi when you hear it the first time. Onyomi sound Chinese, Kunyomi sound Japanese. They're in very different styles. And even if you couldn't, it rapidly becomes apparent due to various other circumstances, such as, well, how they're used. I wrote ショク rather than しょく because I know it's an onyomi. I also know that オン and イン of 音 are on, and おと and ね are kun. I know that クン of 訓 is, ironically, on.
I haven't studied this, beyond reading a wikipedia definition of what 音読み and 訓読み are when I first started, and looking up words in the dictionary
Of course, I'm not saying a study of readings isn't beneficial in some way. But I couldn't say that it's necessary, or that not studying it hampers you in any particular way. Of course, I've never practiced the alternative to an appreciable level...
I can say, however, that this way is incredibly stress-free. I did actually try to learn the traditional way in the very beginning, and I found it absolutely torturous and slow, through about 250 kanji, and I forgot everything constantly. Conversely, learning this way has been only fun, and I don't forget as much. This may be a result of my specific approaches to both styles, but I can only speak to what's true of my personal experiences.