The actor played the vet who put down Chesney's beloved dog Schmeichel last year.
"I went to Manchester and filmed about three episodes of Coronation Street, during which I had to diagnose and put down Schmeichel," he told the Sunday Mercury.
"I know it's only make-believe but there are a lot of people who take Corrie as gospel.
"The day after, I faced some angry jeers and shouts in the street. I'd be walking along, minding my own business, when I'd hear, 'Oi! You're the guy who killed Schmeichel'."
Asbury added that the sad scenes were actually quite humorous to film as the dog was much more active behind the scenes.
"The dog was hilarious. He was supposed to be on his last legs with renal failure or something, but this particular dog was only about a year old and was very lively.
"Every time I said, 'We've got to put him down', he would jump up with his tongue lolling out, looking the picture of health."
The former Royal Shakespeare Company actor recently toured with director Ed Hall's acclaimed Shakespeare company Propeller.