You could argue that music plays a role in any country's stories, but I have come across musical themes a lot in what little modern Japanese literature I've read. I'm reading Kinshu now and Mozart's music plays a role in literal and metaphysical ways. Kira Kira Hikaru's sort of ending theme is Billy Joel's She's Got a Way. Murakami Ryu's Audition features Verdi. There are books whose music references I can't remember because they were Japanese musicians I didn't know about beforehand. I'll write it down next time.
I don't know his books well enough, so I can't name specifics, but I would speculate that you could list one or two thematic songs used in each of Murakami Haruki's books. Please do.

[Zgarbas]

Kafka on the Shore has classical music (maybe Beethoven? I forget the specifics), Norwegian Wood has The Beatles, South of the Border has Nat King Cole (hilariously enough, Nat King Cole never actually played the song South of the Border), IQ84 so far starts with Symphonietta (sp?)... I think there's a song for each novel, but I forgot the rest ^^'

I don't think you actually need to know the songs to catch the reference, though. Most of them are severely taken out of context, if not outright wrong. I am not a fan of Mozart (way too pink) but I still enjoyed Kinshu (which didn't fit my view on Mozart at all) =). The feeling the characters have about the music is more important than the music itself.