Either the 30's, 40's, 50's music is the only music that survived the great war, or people in the radioactive future are very nostalgic and are dreaming of what once was and therefore only playing these tunes.
I love the music in the Fallout games and want more Billie Holiday, on the other hand I do agree that music would have evolved in the years leading up to the Great War. Plus, it's been 200 years, are there no means of making music in the wasteland?
Sure, people have bigger problems than recording tunes, but music is an important part of life and for people to lose that in the 280 years since the war is rather sad. I mean, we had musicians in New Vegas, and we have radio stations... so, why can't some "new" sound spread or be heard? I have no idea what it would be, but the roots should be in the 40's/50's and just adding sounds of available instruments (guitars, steel drums etc.)