The 34-year-old was offered a one-year deal but, having spent the majority of last season as number two to Keiren Westwood, he has opted to seek first-team football elsewhere.
He told the club website: “This has been a really tough decision and one that I have mulled over through the summer but I have decided to move on.
“I want a new challenge at this stage of my career. Everyone loves to play first team football and I am no different.
“I came back to the club this week to do things the right way, which is in person because I have a lot of friends at this football club who deserve full respect.
“I would like to thank all the staff who have been so supportive in my three years with Wednesday and wish the new head coach and the new chairman nothing but the very best.
“And, of course, I would like to thank the fans, who have been different class and I hope to see them at the other end of the field soon.”