No, that sounds like normal English to me. You can put the emphasis on the "or" in speech - I don't drink coffee OR tea.

You could say "I like neither rock music nor chillout music," but that would be very formal. "I don't like either rock music or chillout music" would be fairly emphatic. "I don't like rock music nor chillout music" is unidiomatic in my AE ears, but that doesn't mean it's not used that way in some other variant of English.

Linguists could tell you about "exclusive or" and disjunctions and whatnot.