Greatest 20th century musical influence... African Americans. Yes, it appears to be a rather vague answer, but who can argue with it? Nearly every popular genre of music we listen to today, was influenced by the early African American traditional songs and their rhythmic work cadences. From early blues and jazz, to rock and roll, to modern rap and hip hop, all of it came as a result of this strong cultural influence, and any discussion about the 'greatest influence', would be incomplete without this acknowledgement.
That said, I know the thread seeks to find specific groups or artists, but it is nearly impossible to isolate one in particular. If I had to go with just one group or artist, I would have to say, The Rolling Stones. If for no other reason than their longevity, and having been a major influence for nearly half the century itself. If I had to pick one individual, it would be Elvis Presley. Almost any performing artist of today, will cite Elvis as a huge influence. Although Elvis is not remembered as a great musician, the 'sound' he brought to the forefront was the nucleus and basis for nearly everything we listen to today.
A couple of honorable mentions here, The Beatles and Bob Dylan. Nearly every band, including the aforementioned Rolling Stones, have tried to emulate The Beatles' formula for success. Nearly every songwriter (including Adam Duritz) has wanted to be Bob Dylan.