For many years, the story has been that it was Cockney Rhyming Slang for “dog’s meat.” The august OED, however, throws some cold water on that idea more or less saying that
It has been suggested that this is short for dog’s meat, used as rhyming slang for feet, but there is very little evidence for such a use.
And here’s some more cold water:
In an article about British boxer Ted ‘Kid’ Lewis and his fancy footwork in the ring:
*On [Chicago boxing manager Howard ‘Kid Howard’ Carr’s] arrival, [Carr] was introduced and then remarked of Lewis, “Some dogs you got there, boy.”
“Bloime me, what do you mean dogs?” returned Lewis, not wise to Howard’s slang parlance.
Then [Howard] explained that he meant feet, which didn’t seem at all to affect the Englishman."
–The Post Intelligencer, Seattle, Washington, January 9, 1916
In a satire on weddings called “The June Groom” credited to an “Al Falfa:”
He saunters forth for the execution [i.e., ‘wedding’] with a half gallop and a half pace and is irresponsible for the behavior of his dogs. Dogs mean ‘feet’ in the best of society.
–The Erie Daily Times, Erie, Pennsylvania, June 14, 1917
In this article about some differences in language terms between British and American English (and the idiomatic phrases that troops in both groups would be exchanging ‘over there’):
Perhaps “plates of meat” will never come to mean “feet” to the Yankee soldier, whose word is “dogs.”
–The Daily Missoulian, Missoula, MT, November 24, 1917
But the first recorded attempt to divine the origin of this mysterious phrase (at least that I have found) was this rather fanciful proposal penned by none other than 14-year old Keenan Wynn–yep, the actor and son of Ed Wynn. He had apparently written an essay for his class, which said entitled "The Origin of the Expression ‘My Dogs Are Barking,’ " which was apparently the most common usage of ‘dogs’ meaning ‘feet.’
In 500 AD, the only form of theatricals were Punch and Judy shows, and the only mode of travel was by wagons pulled by huge Saint Bernard dogs. Each show had a complement of five wagons, drawn by four dogs. When they had gone a long distance, the dogs would grow tired. They would bark. This was the signal for their driver to turn around and shout to the other driver, “My dogs are barking.” Then they would all rest. This expression later on came to be used by people whose feet hurt them on long walks.
–The Chicago Daily News, Chicago, Illinois, December 18, 1930
Also in 1923 Babe Ruth was quoted as saying “My dogs are barking a bit.”
–New York American, New York, New York, March 13, 1923
My earlier question on the subject from 2001., which offers Cockney Rhyming as the answer. Didn’t think it was worth bumping, and, anyway, the title was dumb .