From the notes to the Penguin Book (1959):

"This ballad is rather a mystery.  Is it an unusually coherent folklore survival of the ancient myth of the slain and resurrected Corn-God, or is it the creation of an antiquarian revivalist, which has passed into the popular currency and become "folklorized"?  It is in any case an old song, of which an elaborate form was printed in the reign of James I.  It was widespread over the English and Scottish countryside, and Burns re-wrote a well-known version.  During the present century, versions have been collected in Sussex (FSJ vol.I [issue 3] p.81), Hampshire (FSJ vol.III [issue 13] pp.255-6), Surrey (FSJ vol.VI [issue 21] pp.27-8), Somerset (Folk Songs From Somerset, Cecil Sharp, 1904-9, vol.III p.9 and vol.IV p.32) and Wiltshire (Folk Songs of the Upper Thames, Alfred Williams, 1923, p.246).  The tune is a variant of that usually associated with the carol,  Dives and Lazarus."  -R.V.W./A.L.L.

This version was collected by Cecil Sharp from "Shepherd" Haden of Bampton, Oxfordshire, in 1909.  It was first published in the Folk Song Journal, vol.VIII [issue 31] p.41.

Other versions on the DT:

John Barleycorn  Robert Burns' version; tune specified as Lull me beyond thee ¹  but not given.

John Barleycorn  No source or tune are given.  It is, however, a (not quite accurate) set of the  John Barleycorn's a Hero Bold ² text commonly found in 19th. century English broadsides.  A traditional version, very close to the text given here but with an additional verse, was collected, with tune, by Bob Copper from John Attrill of Fittleworth in Sussex, in 1954, and published in Folksongs of Britain and Ireland (Kennedy, 1975).  A midi of the melody goes to Alan's Mudcat Midi site.

John Barleycorn  No source is specified, and the tune given -more or less the Penguin version- doesn't fit the text, which bears a certain resemblance to the 1972 recording made by Steeleye Span; this was an arrangement of a version collected by Fred Hamer from Billy Bartle of Bedfordshire.

John Barleycorn My Jo  A temperance song using the same metaphor, based on  John Anderson My Jo,  with tune.  From the Grieg-Duncan Folksong Collection.

Curiously, there is a stray soundfile in the Download version of the DT, JBARLEY3, which has embedded lyrics that don't match any of the texts.  The tune does bear a vague resemblance to Lull me beyond thee, though.  Perhaps Dick can solve the problem when he gets back from England.

@drink

There is an entry at  The Traditional Ballad Index:
John Barleycorn  

Bruce Olson has the text of a 16th century Scots version from the Bannatyne MS at his website,  Roots of Folk: Old English, Scots, and Irish Songs and Tunes:
Sir John Barleycorn / Allan-a-mault  

There are some early texts at Steven Earnshaw's site;  Altered State: England, Literature, and the Pub.  Not given elsewhere here are:

The Ballad of Sir John Barley-corn   ("As I went through the North Country")

...together with an early text on much the same subject,   Exeter Riddle 'Ale'  

As might be expected, there are a quite a few broadside versions of this song at  Bodleian Library Broadside Ballads.  Here is a selection of the easily-legible ones:

² John Barleycorn's a Hero Bold:

John Barleycorn  Printed in 1859; printer unknown.

John Barleycorn  Printed between 1863 and 1885 by H.P. Such, Machine Printer and Publisher, 177, Union Street, Boro', S.E. London

John Barleycorn  Printed between 1858 and 1885 at the "Catnach Press," by W. Fortey, Monmouth Court, Bloomsbury London

John Barleycorn  Printed between 1840 and 1866 by J. Harkness, Preston.

John Barleycorn  Printed between 1863 and 1885 by H. Such, Machine Printer and Publisher, 177, Union St., Boro'. S.E. London.

John Barleycorn  Printed between 1860 and 1883 by H. Disley, Printer, 57, High Street, St. Giles London

Sir John Barleycorn:

Sir John Barleycorn  Printed between 1849 and 1862 by H. Such, Such's Song Mart, 123, Union Street, Borough, London.

Sir John Barleycorn  Printed between 1849 and 1862 by H. Such, London.

Sir John Barleycorn  Printer and date unknown.

A pleasant new ballad to sing ev'ning and morn, of the bloody murder of sir John Barley corn   To the tune of: Shall I lye beyond thee ¹  Printer and date unknown.

These are large images.

¹ This tune (also called Lull[e] Me Beyond Thee) may be found at Bruce Olson's   Tunes for 16 and 17th Century Broadside Ballads.  For myself, I do not see how Burns' text could comfortably be sung to it, though two relatives of the tune, "Stingo" and "Up in the Morning Early" could be made to fit easily enough.  Perhaps someone could elucidate.

X:284
T:B284- Lie Lulling Beyond Thee
Q:120
L:1/4
M:6/4
K:Am
E2Ec3/2 B/c|d3/2 c/de3|E2Ec3/2 B/A|^G3A3:|\
c2cd3/2 c/d|e3/2 f/ed2G|c2cd3/2 c/d|e3d3|\
e3/2 f/ed3/2 c/B|c3/2 B/Ae3|E2Ec3/2 B/A|^G3A3|]

There is some discussion of the evolution of this tune from Stingo (John Playford, 1650), through Lulle me beyond thee and Up in the Morning Early (Select Songs of Scotland, Gall and Inglis) to the Shetland fiddle tune Sister Jean, at Jack Campin's  Scales and Modes in Scottish Traditional Music.

The Winning of the Mead in the Icelandic Prose Edda (Snorri Sturluson, c.1200) offers an interesting parallel to John Barleycorn.  When the two tribes of gods, the Æsir and the Vanir, made peace between each other, they all spat into a vessel and made from the contents the god Kvasr, who was able to answer all questions.  Kvasr was later murdered by two dwarfs, who fermented his blood with honey in three vats, to make the Mead of Inspiration.  This was subsequently taken from them by the giant Suttung, from whom Óðinn reclaimed it through a mixture of trickery and the seduction of the giant's daughter.  Drinking the entire contents of the three vats, he escaped in the form of an eagle and returned the mead to Ásgarðr.  A little leaked out on the way, falling to earth, where it bestowed a scattering of Poetry upon Humankind.  Kvas was "the word for strong beer used by the eastern neighbours of the Germans, and [is] still used in Jutland for crushed fruit". (Gods and Myths of Northern Europe, H.R. Ellis Davidson, 1964.)

Malcolm