Pretty much the entirety of the Cocteau Twins discography is this. A few Enya songs as well.

Also: "Bork, bork, bork!" http://www.youtube.com/watch?v=sY_Yf4zz-yo

EDIT: Let's be honest here, quite a few ordinary pop songs are also gibberish masquerading as coherent language.

EDIT: Richard Feynman even got into the act once as a young man. Ironically, he was an English speaker doing gibberish Italian: http://www.bookrags.com/studyguide-surely-youre-joking-mr-fe...