The Scotland international, 29, has been at Turf Moor since January 2007 but has struggled with injuries for much of the season just ended.
He told the club's official website: "It's the end of a journey that I've thoroughly enjoyed. It's been an exhilarating experience to get to the Premier League, to win the play-off final at Wembley and to lead that bunch of men like I have donehas been an honour and a privilege.
"There are many great people at Burnley that are the heart and soul of the club. The supporters have been fantastic, especially this season. They've always backed us and they've given us that extra bit of support when we needed it. I'm a bit sad to go, but I'm looking forward to my next challenge."
Left-back Stephen Jordan, goalkeeper Nicky Weaver and midfielder Adam Kay have also been allowed to leave Turf Moor.
Keeper Brian Jensen, defender Michael Duff and forwards Robbie Blake and Steven Thompson have all be offered fresh terms and Graham Alexander and Clarke Carlisle have penned new two-year deals.