The 30-year-old joined the Clarets from Derby in 2018 and made 82 league appearances for the club.
Vydra's contract expired at the end of June while he was recovering from a cruciate knee ligament injury which sidelined him for the last four matches of the season.
The forward said Burnley had only made him one offer of fresh terms, which he could not accept.
"I would like to say farewell to my amazing fans who have supported me throughout my time here in Burnley FC," Vydra said on his Instagram account.
"I've been lucky to experience the family atmosphere of the club, great teammates and outstanding support from our doctors and physiotherapists that have helped me with my injury.
"I've done my very best to play to my fullest potential during the length of my contract, not afraid to risk getting injured even towards the end of this contract.
"That's just who I am and I wanted to help my club at any cost. Unfortunately during a match against Watford I got seriously injured and the club got relegated to the Championship.
"Yesterday I went to see my brilliant surgeon Andy Williams, who was very pleased with my recovery. I'm ahead of the estimated recovery time and I was told I could soon be on the field again.
"Negotiations with the club were minimal, I received one offer, which I could not accept.
"To the club and to the fans, I wish a great season and the best of luck."