Mancini, who was sacked earlier this month, often appeared to have issues with Richards and regularly questioned his fitness or form, or used him sparingly.
But the right-back has spoken in glowing terms of the Italian, who oversaw FA Cup and Barclays Premier League victories in 2011 and 2012 respectively.
Richards, speaking on the club's post-season tour of the United States, said: "Mancini did fantastic in the three years he was here.
"The owners have decided they wanted a change and that was up to them but I'd like to thank Mancini for what he did for me - winning the Premier League, FA Cup and Community Shield.
"It was brilliant, I'd never won anything before.
"I'm happy for that but we are moving forward now, and looking forward to whoever it is going to be and seeing their plans."