The 21-year-old, who made just two league appearances for Roberto Mancini's side, leaves the club for an undisclosed fee after agreeing personal terms and signing a four-year deal with the Robins.
The Championship outfit had been on the lookout for a new left-back following the departure of Jamie McAllister after six seasons with the club, while Dundee United defender Paul Dixon had also been linked, but boss Derek McInnes is pleased to land his target.
He told the club's official website: "The left-back position is one that we've needed to strengthen and fill.
"Greg is someone who has interested us for a long time. We scouted Nottingham Forest a number of times at the back end of last season and he really caught the eye.
"He's a good player who is only going to improve. Greg is a solid defender, determined and full of energy and pace.
"We had a number of options put to us in that position but good left-backs are hard to find and Greg was always the one we wanted. I believe we have secured the services of one of the best up-and-coming left-backs.
"It's a great financial deal for the club and we get an excellent player out of it on a long-term, secure contract."Cunningham has spent time away from the Etihad on loan in the past few seasons, impressing during spells at Leicester City and Nottingham Forest.
In 2010, the Galway-born defender received his first senior international call-up under Giovanni Trapattoni and made his debut in the May friendly against Algeria at the RDS.