Moore, who was selected in the third round of the 2013 draft, started 22 games the past two seasons and has 7.5 sacks in his career.
Moore tweeted about his emotions.
This has been more than a home to me... So much if my heart is in the bay... My teammates RaiderNation Yall are family to me. Will miss Yall— Sio Moore (@MrOakTown55) September 4, 2015
I wish I could've of did it for yall... I do. I gave yall all I could. Love yall ✊#55Strong— Sio Moore (@MrOakTown55) September 4, 2015
Moore fell out of favor with the coaching staff in Oakland. He missed offseason workouts and part of training camp after having hip surgery in January, and he mainly played on special teams during the preseason. He played only nine snaps in Oakland's preseason finale at Seattle.
Moore joins a linebacker group with the Colts that features D'Qwell Jackson, Jerrell Freeman, Erik Walden and Trent Cole. Colts coach Chuck Pagano said Friday that linebacker Robert Mathis might play in their Week 1 game at Buffalo.
ESPN NFL Nation Raiders reporter Bill Williamson and The Associated Press contributed to this report.