Kingdom Hearts isn't a terrible story, but it does have a little more than its fair share of cheese.
I enjoyed 358/2 Days' story more than I thought I would. Its lack of Disney character interaction made it the least cheesiest story to date. I know there are some people who would disagree, but getting rid of the Disney parts of the story would be the best choice for the franchise. Donald, Goofy, and Mickey play their parts very well, but the lack of character depth in any other Disney character is just forgettable. Its like there was no point to include them in the game anyways.
I'm excited about Birth by Sleep. It might be able to justify the horrible story transition between KH1 and 2.