In addition to the interesting and sometimes surprising recommendations (Gramophone December 2012) of Peter Dickinson, who is assuredly an expert on American music , I wish to add a few. Randall Thompson's Second Symphony (1931), quintessentially American in its rhythms, is best represented by Leonard Bernstein's Sony recording 60594. On the same recording, with the New York Philharmonic, in the Bernstein Century series, is David Diamond's appealing Fourth Symphony (1945) and William Schuman's Third Symphony (1946), all at a bargain price these days, at least at Arkivmusic.com.
My own nomination for the greatest American Symphony is Harold Shapero's Symphony for Classical Orchestra (1947), particularly the recording by Bernstain, also on Sony's Bernstein Century series, Sony 60725. I have found this work exciting for half a century. Andre Previn has a more recent recording but in my opinion Bernstein brings out the forward motion and driving rhythms better.
This is a large-scale, four movement work, interestingly influenced by both Beethoven and Stravinsky, although too much has been made of these connections; neither of those composers could have composed the work, which is in the composer's own voice.