In a statement on Rovers' official website Williams said: "We have enjoyed a first-class, mutually beneficial relationship with Mark.
"He leaves the club in a strong position with three consecutive top-10 finishes and an excellent squad of players.
"In turn, we have helped Mark to establish himself as one of the most sought-after young British managers in the business, having previously had no club management experience.
"We wish him well. We would have liked him to stay on, but he made it clear he wanted a fresh challenge.
"Now our focus switches to the search for a successor. We have enjoyed eight years of success and stability with Graeme Souness and Mark, and now it is important that we build on our strengths.
"We must make sure it is three successful appointments in a row."