Newbee, I second Bernstein as one of the most influential American musicians. He really brought the idea of classical music to the masses. His music also had very wide appeal after his "early" days.
Vegasears, I too agree that Copland was an excellent American musician. He was able to capture the American spirit and put it to music. His pieces, I believe will far outlive us all as a look into what life was like in the United States in the 20th century.
I think that Arthur Sullivan is perhaps the most influential American Musician. He along with William Gilbert gave birth to the ultimate American art form, the musical. I think that the musical really is the United States' most important contribution to world music...even though I am not a fan of the musical, together the musicals in production today make up most of the live music people pay to see...
These are the classical musicians. There are many others that have had huge contributions to popular music...perhaps too many to name. It is interesting to see Grand Master Flash being mentioned in this thread...It is true that he gave birth to the highest grossing form of music in all history...Rap...I, for one, have listened to my fair share, hard not too growing up in the 90's, but it sounds like s*#t on my hi-fi setup.
As for artists that will be influential going into the 21st century. I think that Dave Mathews, and his eclectic approach to the band has created a sound that appeals to many people just because it has depth in todays depthless society...plus it has wide spread appeal. As for classical musicians that have a lot of influence going into the 21st century. I would include John Williams, Danny Elfman, and Philip Glass, their film scores are the newest, next huge contribution to classical music. The problem is that there is much more money for these modern composers in music for film than in the old method of writing symphonies...I for one look forward to what happened to music in the 21st century...should be interesting.