Rodriguez was picked up in May on charges of DUI, speeding and an improper lane change. In March, Rodriguez -- the Bears' fourth-round pick in 2012 -- was arrested in Miami and charged with resisting an officer without violence and disorderly intoxication, charges that eventually were dropped.
Rodriguez was told his release wasn't for on-field reasons, the source told Rapoport.
In a related move, the Bearssigned fullback Tony Fiammetta on Monday, according to his agent, Dennis Boyev. Fiammetta comes to the Bears after stints with the Carolina Panthers, Dallas Cowboys and New England Patriots. He'll compete with Harvey Unga for snaps, but Fiammetta hasn't appeared in a game since 2011.