Neat idea. This is something I don't generally include.
I think rather than trying to pick "real world" genres for fantasy races, I'll try to suppose/imagine the sounds/instruments they would use.
Humans: basically...anything. harps/lyres, mandolins, drums, tamborines, various brass/horns, some violin/fiddle in certain regions, harpsichord or pipe organ (if you have a civilization that advanced). I would say the style of music for a given human community would differ from region to region and is influenced by the other races around/within it...and probably multiple styles depending on the size of the community.
Elves: harps/lyres, lots of pipes/flutes, lots of reeds/woodwinds, chimes, bells, possibly piano. Very little or subtle use of brass instruments (French Horns strike me as a very "elf" kind of brass instrument, but a Trumpet or Trombone for example would be far to...harsh/brash). Lots of use of voice/complex choral harmonies. Little bass, lots of intricate and melodic treble.
Dwarves: Lots of drums/percussion, lots of brass/horns. I think lots of "deep" sounds. Tubas/Sousaphones (sp?), Bassoons,...I like the bagpipe idea someone mentioned above...not because it's Scottish, haha. But just that I imagine dwarves would have some type of instrument like that that no other race would have figured out how to make and would be distinctive to their culture. Lots of somber sounding music. Lots of bass, little treble, long "slow" sounding music...lots of "marches" type music...Chanting type "songs." I imagine Gregorian-style chants (but not so much on the treble side) during working or marching to war...AND upbeat rousing songs for drinking and/or victories.
Gnomes: Hmmm...LOTS of fiddle/violin and strings in general, lots of pipes and woodwinds. Maybe bagpipes go better here...hmm. I'm picturing lots of upbeat "high-stepping" dancing songs. A typical gnomish quartet would be someone on a flute or clarinet-like pipe, a fiddler, someone on an upright (gnome sized) bass, and percussionist (with a wash-tub drum, tamborine and triangle). I'm thinking like traditional Irish pub and step-dance music.
Halflings: Very similar to gnomes for "party music/drinking songs" but also more refined sounding music. Lots of strings here, violins, cellos, guitars. I might say that the keyboard/piano might have started in the comfortable burrow parlors of halfling homes? Halflings, like elves, also very much enjoy singing and songs with their music, something dwarves and gnomes are less likely to do.
Orcs: I agree with the poster who mentioned beating longs and drums. I concur with "the louder the better." Wild banging, booming and screeching sounds.
...do with it what you will.
Thanks for making me think about something I haven't before.
--Steel Dragons