Bryan was a consensus All-American at the University of Oklahoma in 1982 and 1983. He was Big Eight Defensive Player of the Year in 1982 and Academic All-Conference from 1981 to 1983. He owns the school record for career tackles by a defensive lineman (365), ranking him ahead of Kevin Murphy and Lee Roy Selmon. He is OU’s eighth leading career tackler regardless of position. “No one outworked him,” said OU coach Barry Switzer.
Bryan was selected by the Atlanta Falcons with the 9th overall pick in the 1984 NFL Draft. Bryan was the NFC Defensive Rookie of the Year in 1984. He spent all nine years of his professional career with the Atlanta Falcons, and still has the tenth most sacks in a career in Falcons' history, with the eighth most yardage lost. He eventually was forced into retirement by a number of injuries, including neck and back pains and a popped Achilles tendon. He was diagnosed with a spinal nerve injury in 1989, but kept playing and started sixteen games in each of the next two seasons. "Rick was a warrior. That's what made him such a good football player," said Falcons All-Pro tackle Mike Kenn when Bryan retired.
Bryan died on July 25, 2009, at the age of 47, in his home in Coweta. He had suffered from congestive heart failure and his family said he died of a heart attack. 2,500 people attended his funeral in Coweta on July 30, 2009.He now has a sign dedicated to him on the road outside the Coweta High School.