The 31-year-old joined the Championship outfit from Marseille on loan in January 2009 and made his move permanent that summer.
"I want to say thank you to the club and to the fans," Givet told the club's official website. "Before Blackburn, I enjoyed my football in Monaco - I played in the Champions League final - but my first three years in the Premier League with Blackburn were the best moments of my career.
"It was an amazing experience for me and one I will never forget. Blackburn Rovers will always have a big place in my heart and I am sad to be leaving, but last season was hard for me and I think it is better for me and my family to go back to France.
"But once again, I say thank you to the club for giving me a chance to play in England and thank you to the fans for your support - I will never forget it."
Givet featured in 133 matches for Blackburn during his time at Ewood Park.