The 31-year-old, who has one year remaining on his deal, is free to search for a new club following two seasons with the Rhinos.
"It was always an ambition of mine to play for the Rhinos first team having started my career at Headingley Carnegie and I am pleased to have done that," Moore told the club's official website. "It has been a great experience for me and I am looking forward to the next chapter in my career next season."
Head coach Brian McDermott added: "Richard leaves the Rhinos with the best wishes of everyone at the club.
"He has been a great servant to Leeds over the last two seasons, always prepared to help out the team wherever possible and having made 16 appearances in both seasons it shows the contribution he has made."
Moore made 32 appearances for the club, scoring one try in the process.