"Green, Green" is a 1963 folk-style song by folk revival group The New Christy Minstrels. From the original song, it has transformed in unusual ways.
I first heard it by Johnny Rivers on his 1965 album Johnny Rivers Rocks The Folk, and then later heard it performed by The New Christy Minstrels. The song is a lyrical tribute to rootlessness based on the proverb "The grass is always greener on the other side" where the singer talks about being a drifting, ramblin' man, a staple of country and folk music. In the more innocent year of 1963, perhaps it wasn't meant to explicitly refer to a man abandoning a woman, but that certainly is one way to look at it. The New Christy Minstrels' version, like much folk music, sounds affected to me. The Johnny Rivers version is a great combination of swinging earnestness and sly irony.
The song transformed even more when it was adopted by the Japanese interstitial television program Minna no Uta into a Japanese version, sung by children. Apparently, it was popular enough in Japan that it might have influenced Koji Kondo, who seems to have reused a short part of the melody for the overworld theme of Super Mario World. (This isn't to say that it was either intentional or extensive enough to be "plagiarism", but I can clearly hear the first four notes every time I hear the Mario overworld song).
I enjoy the song, and its unusual history is also interesting.