The 25-year-old, who was in the last year of his contract, has penned a three-year deal with Boro after three years at Ashton Gate.
Managing director Jon Lansdown told the club's official website:
"Albert has been a joy to watch and a pleasure to work with over the past three years.
"His level of professionalism and courtesy is a credit to himself. I personally wish him all the best for the future and hope to see him go on to play in the Premier League."
Head coach Sean O'Driscoll added: "We are sorry to see Albert go, but his departure is not a surprise and we have been planning for it all summer."
Adomah made 136 appearances for City, scoring 17 times following his arrival from Barnet for £140,000.