Rodriguez, a fourth-round draft pick out of Temple in 2012, was ticketed by Illinois State Police on May 31 for driving under the influence, speeding and improper lane change.
Rodriguez was also arrested in Miami Beach, Fla., in March for resisting an officer without violence and disorderly intoxication after a car in which he was a passenger was involved in a traffic incident. Those charges were eventually dropped.
Bears general manager Phil Emery said he was "extremely disappointed" in Rodriguez during OTAs last week.
"Evan is a very likeable guy," Emery said Tuesday. "We have talked. Disappointed. I would use the word extremely. And we'll leave it at that.
"There are a lot of things we have to work through. All off-the-field incidents are subject to NFL policies and guidelines. We've got to work through these things."
Rodriguez appeared in 12 games as rookie and started five, contributing four catches for 21 yards. During voluntary minicamp and organized team activities, the Bears worked Rodriguez at fullback, but he figured to play somewhat of a hybrid role as a blocker out of the backfield and as a tight end.
The Bears also signed veteran fullback Tony Fiammetta and receivers Devin Aromashodu and Jerrell Jackson. Fiammetta and Aromashodu were part of the large group of free agent players that attended a tryout held by the Bears last Friday at Halas Hall.
Aromashodu returns to the Bears after spending the past two seasons with the Minnesota Vikings. He had 34 receptions for 447 yards and four touchdowns in 24 games with the Bears from 2009-10.
Jackson played one game as a rookie with the Jacksonville Jaguars last season.