This has been partly forgotten nowadays, though, and since Tolkien wasn't interested in The Arabian Nights (at least not as far as I can tell; he certainly doesn't seem to have incorporated any themes or ideas from them in his work), the influence of the Arabian Nights "vibe" has gradually faded from a lot of fantasy over the last few years.
However, with even a cursory look at writers like Clark Ashton Smith, Abraham Merritt, and especially Robert E. Howard, shows that the Arabian Nights were an enormously influential and important influence on Sword & sorcery from the very beginning.
There are reasons to believe that the Arabian Nights influence is still with us, of course. The upcoming release of Disney's Prince of Persia (not to mention the continued success of the game series on which its based) will be, I hope, nice fodder for fantasy fans. Paizo, proving as always, that they are tapped into what makes fantasy tick, has released several products that have an Arabian Nights tone and feel, especially Qadira but also Katapesh and the entire Legacy of Fire adventure path. During the second edition era we got the Al Qadim setting, which was more successful than TSR had expected it to be (and was therefore extended an additional year and more products were scheduled and released before the line ended.)
Anyway, I bring this up not just because I just bought the recently released mp3 download from Amazon of the movie cast soundtrack to Kismet (although I did) but also because I've really loved the Arabian Nights vibe in fantasy. I think it's an integral part of the genre as I know it, and I want to see it preserved and even actively encouraged.
Oddly since it has nothing to do with the Arabian Nights or anything, the music for Kismet is heavily based around classical music by Georgian/Russian composer Alexandr Borodin, which was re-arranged and layered with vocals and a few jazzy touches. There are parts of the soundtrack, however, which remain untouched Borodin; from his String Quartet No. 2, In the Steppes of Central Asia, Prince Igor and other works. Rimski-Korsakov's symphony based on Scheherazade was not used.
There's your pop culture trivia of the day.