Mancini was sacked earlier in the month and appeared to have a shaky relationship with Richards at times during his spell in charge, publicly questioning the full-back's fitness and form on occasion.
But Richards has praised his former boss, who led the club to FA Cup glory in 2011 and a Premier League title win in 2012.
"Mancini did fantastic in the three years he was here," said Richards.
"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 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 (who takes charge) and seeing their plans."
Players to back new man
Manuel Pelligrini is leaving Malaga at the end of the Spanish season and is expected to replace Mancini at City.
And Richards added: "That's in the papers and what everyone's speaking about.
"But you never know, sometimes late runners come in. Whoever it is, we will give 100 per cent and we'll be fully behind him."
Richards was speaking during City's post-season tour of the United States and the team have arrived in New York for a second friendly against Chelsea.
"The time in New York has been fantastic," said Richards.
"It's probably one of the best places on Earth and I'd like to come back again.
"We are trying to expand, we are trying to get fans worldwide, so coming to America is perfect for us.
"MLS is getting stronger every year and we believe coming here and trying to expand it is only right for the club."