The 25-year-old completed his move to the Riverside Stadium on Thursday for an undisclosed fee.
Adomah has revealed that Mowbray had pursued him for two years before finally securing his signature.
"He's been keeping tabs on me for the past two years," Adomah told BBC Tees.
"Eventually he did get me so I have to do well for him now. Hopefully I'll provide for him. It puts a bit of pressure on but, when a manager wants and signs you, that means he's up for you."
Adomah scored seven goals in 46 appearances for Bristol City last season.