Wow, that leaves a lot of lattitude!
Hmmm, how about:
Sharara, Asha Bosle (in Hindi)
Take Me With You, Prince (even though it's Prince, it's not dirty/vulgar)
One Big Love, Patty Griffin -- covered by many people
Love Shack, B-52s (Ok, maybe that's not super-romantic, just fun)
I'd Do Anything to Turn You On, Roxy Music
Falling Slowly, Glen Hansard, from the Once soundtrack -- might be more serious than you're looking for. It sure is romantic, though!
You Send Me, Sam Cooke
Ob La Di, the Beatles
Crimson and Clover, Tommy James or any of a bunch of excellent covers
LOTS of old jazz tunes -- The Way You Look Tonight springs to mind, but really too many to choose from!