I found out the answer to my own question, but thought I'd post here in case anyone else is interested. The song is 'Cheyenne' (or Shy Ann) and it's much older than I thought, being from 1906. I was hoping it might be a song from the score of a 1930s Western and although that's not where the song originates, according to Wikipedia, "the song became a staple of the underscore of western films, to the point of being stereotyped".
So far the only recordings of it I've found are over 100 years old, but I'm hoping to find a version that sounds more typically Western or cowboyish, if anyone can help.