Atlanta announced the Alford move Tuesday. Bryant posted a message about his release on social media on Wednesday afternoon.
"I was informed last night that the team was moving on from my services and that I would be released," Bryant wrote. "I want to say thank you to all the fans that have stood with me and my family during our time here. I take great pride in my body of work on and off the field. I hope I have represented you well while I was here.
"It was an honor to be a part of some big moments in this franchise's history. This chapter is now closed. I look forward to bringing the success and consistency that I've produced to my next team."
Bryant, 43, made 95.2 percent of his kicks in 13 games last season for the Falcons. The 18-year veteran has made 82.2 percent of his kicks during his NFL tenure. He has made 98.8 percent of his extra points. Bryant made the Pro Bowl in 2016 after making an NFL-high 56 extra points. He led the league by making 93.1 percent of his field goals in 2011.
He began his career in 2002 with the New York Giants. He also played for the Indianapolis Colts and Miami Dolphins before landing with the Tampa Bay Buccaneers in 2005. Bryant joined the Falcons in 2009.
Alford also posted about his release on social media. The second round pick in the 2013 NFL Draft has played his entire career with the Falcons. Alford had 50 tackles and 11 passes defensed in 15 starts last season.
"We thank Robert for all that he brought to the team during his six seasons in Atlanta," Falcons general manager Thomas Dimitroff told AtlantaFalcons.com. "He was a significant contributor to our defense and we have a great deal of respect for him. We wish Robert all the best going forward."