Dyche was sacked and replaced by David Moyes.
He has now released a statement through the League Managers' Association, saying: "It was an honour to manage Everton, a football club with a significant heritage and an enormous following in Liverpool and all over the world, through one of its toughest periods in its history.
"Despite the challenges we faced, I wanted to ensure that the club’s narrative was focused on the positive direction it will take in the future and that the team could focus on competing in the Premier League in the present.
“I thank my staff, the players and all involved who played their part, as it simply wasn’t possible without their support and expertise. I also thank the fans who came through for us many times when we all needed their support.
“The right time has come to leave the club, with the team in good shape. David is a manager who I have great respect for and I believe he and his staff will help to push the club forward again and build on the stable footing we have implemented in the past two years.
“I wish everyone well going forwards including the new owners, who I hope will take the club to the next level and deliver the success which every Evertonian wants.
“My part as a custodian has been played and I will forever take great pride in that. After going through some extremely difficult situations, I never lost faith in everyone around me and we did everything we could to protect the badge, safeguarding it and the name of Everton Football Club.”