Actually, all the non-Disney themed worlds had the storyline. Think about it.
Destiny Islands- Where Sora lives, was born, and where the storyline starts.
Traverse Town- First new world to Sora and where Sora learns about heartless keyblades and such.
Hallow Bastion- Where all the baddies are, and where we first see Ansem. Also where the Princesses were held, where Sora turned into a Heartless, and many other events.
The End of the World- The world of the Heartless. Ansem was defeated there, and it was the last world. That was also the source of Kingdom Hearts.
Twilight Town- The first world in KH2, and where the first discoveries of Nobodies are made. Also Roxas lived there (fangirls screaming in the background).
Radiant Garden (HB2)- Where a lot of the storyline takes place, and the place where many secrets unfold.
The World That Never Was- The Organization world, and the final world of KH2.
The only exception (but a very small one), was with Disney Castle. But it had nothing to do with the storyline of the Organization. It simply was where all the action with Pete and Malificent was. Also, we were all so confused when we couldn't go to that world in the first KH.