I first heard this from Woody Guthrie, but that just tells you where and when I started listening to southern rural music, not where the song comes from. If we were looking for the earliest form of blues, this song is as good a nominee as any, and by the turn of the twentieth century it seems to have been known all over the South.
It was first recorded by a Virginia singer named Henry Whitter late in 1923, but a version that sounds even older was done the next year by a singer and multi-instrumentalist from North Carolina named Samantha Bumgarner, who traveled to New York in 1924 and made a dozen sides on fiddle and banjo — by some reports the first southern mountain banjo recordings ever made. She called her version “The Worried Blues,” and it sounds to me like the sort of music that was played on banjo before guitars became common in the southern mountains, mostly by African American musicians. Bumgarner was born in 1878, and by the 1920s was considered a representative of older and potentially dying traditions. She performed every year at Bascom Lamar Lunsford’s folk music and dance festival in Asheville, and was one of the players who inspired a teenage Pete Seeger to take up the five-string banjo after his father brought him there in 1935.
This song is typical of the sort of proto-blues (clearly related to later blues, though not yet called by that name) that had been common in black communities in the later 19th century, but had largely fallen out of favor by the time recording arrived and is mostly known from the work of white singers — one of the quirks of American musical history is that, at least until the later 20th century, African Americans were rarely nostalgic for any “good old days,” while white southerners were deeply devoted to their region’s past, and as a result white artists often preserved archaic black styles.
Whitter’s version is a lot closer to Woody’s, and may well have been Woody’s source, since it was a very popular record. They sing a lot of the same verses, which is a pretty fair clue, since the form of the song is so simple that there were hundreds of verses circulating and people routinely made up new ones. Some of my favorites are from the versions by Jim and John Jackson (unrelated, and recording forty years apart), titled “Going Down to Georgia on a Hog”… but I still sing it more or less like I learned it from Woody.