The Mario Kart Series consists of 8 parts as far as I read so far. Two of them are Arcade Games, of which I dont know much. The others all do have (at least) one track named Rainbow Road. So I thought, maybe they have some parts of the background music in common. Which appears not to be the case. Actually, they seem all to have different background melodies. That is strange. Anyway, I have compiled the following youtube-links with the melodies.
The first is from the original Super Mario Kart from 1992. I consider this being the best one, because to me it is the only one that sounds as if it was composed with at least some kind of inspiration:
The second one is from Mario Kart 64 from 1996. It is really strange that they have not chosen to take the known melody and remix it somehow instead of making a completely new theme. But if only they would have made it good:
Mario Kart Super Circuit, from 2001, for the GBA, has two courses, one is a new rainbow road track, the other one is similar to the one from Super Mario Kart. The music of the new one is a new melody that at least features a few motives of the melody from Super Mario Kart:
The music of the one similar to the SNES is almost the same as the original melody:
The game for the GameCube, Mario Kart Double Dash, from 2003, also has another melody:
Mario Kart DS is made for Nintendo DS, and also has another melody:
The last Mario Kart so far is the one for the Wii, which has also a melody, and sort of the worst: