"Yankee Doodle" is probably the most well known of all tunes to come out of the Revolutionary War. It was originally a British song, mocking the Continentals, but it had a catchy melody, and was quickly taken over by the Americans as a rallying tune. Here is a video of the Towpath Fife and Drum Corps performing "Yankee Doodle" on parade.
I also came across this clip of "Yankee Doodle" sung by folksinger Charlie Zahm, which I thought was rather good as well. It is, perhaps, not entirely historically accurate, what with the bodhran and all, but it makes for a stirring arrangement.