The 43-year-old, who made over 200 appearances for the club, was the oldest player to play in the Football League last season.
"He's gone with everyone's best wishes," Dickov told BBC Radio Sheffield.
"We wanted him to stay on as a keeper and in an ambassadorial role, but he has had an offer from elsewhere he felt he couldn't turn down.
"I told him I was looking to bring in a number one goalkeeper and he totally respected that. He's been a fantastic servant to the club and a brilliant professional."
The Scottish international made his name at Wimbledon where he spent six seasons as a Premier League regular before top-flight spells with Tottenham and Chelsea.
He joined Doncaster from Leeds, initially on loan in 2006, and would help the club clinch promotion to the Championship with a play-off victory against the Elland Road outfit.
Last season he spent a spell on loan with AFC Wimbledon before returning to the Keepmoat to help Rovers clinch a return to the second tier.