I talked about this a bit in my Master's thesis last year, and I don't have a whole lot to add to what's already been said here. It's not so much that supernatural elements are gone from American songs, as it's already been pointed out that plenty of old and new songs still have those elements.

What is pretty evident in the study of Anglo-American ballads, and murder ballads in particular, is that the Americanized versions of particular songs tended to lose the ghosts, presumably because people either a) found belief in the supernatural to be outdated, or b) religious revivals in the colonies led people to change those songs accordingly. Murder ballads originating in America tend to leave off the ghosts and deal with retribution instead. It's just the folk process at work.

The quote Masato posted from Tristram Coffin explains it pretty well, I think.