The deal was first reported by ESPN NFL Insider Adam Schefter.
Podlesh will take over the team's punting duties after the Bears parted company with veteran Brad Maynard.
Originally selected by the Jacksonville Jaguars in the fourth round of the 2007 NFL draft, Podlesh was voted a Pro Bowl alternate last season. He posted career highs in average yards per kick (43.8) and net average (39.2).
"[The Bears] seemed right out of the gate to be a good fit for him," agent Richard Rosa told ESPNChicago.com. "He had multiple teams after him right out of the shoot, but [the Bears] were very aggressive, and Adam felt really good about the commitment to special teams in Chicago."
In his four-year career in Jacksonville, Podlesh dropped 75 punts inside the 20-yard line -- 26 of them in 2010 -- an important trait to possess in the Bears' directional punting scheme.
Maynard had said on ESPN 1000 that his deteriorating relationship with special teams coach Dave Toub contributed to his departure.
The Bears countered that the decision was based on Maynard's 2010 numbers in which he averaged 40.1 yards per kick to go along with a net average of 35.2 yards. He did place 24 punts inside the 20-yard line last year.
Jeff Dickerson covers the Bears for ESPNChicago.com and ESPN 1000.