Rodriguez was designated for assignment on June 4. The 26-year-old pitcher went 0-1 with a 4.00 ERA in 17 relief appearances for Washington this season.
Rodriguez has spent all or part of five seasons in the major leagues with Oakland (2009-10) and the Nationals (2011-13). He set a career high with nine saves for Washington in 2012.
Dickson, 22, was 2-2 with a 6.88 ERA at Class-A Kane County. He originally was drafted by the Cubs in the 35th round of the 2011 draft.
The Cubs designated right-hander Eduardo Sanchez for assignment to clear a roster spot on the 40-man roster for Rodriguez. Sanchez, 24, was 0-1 with a 9.82 ERA in six relief appearances for Triple-A Iowa. Sanchez was acquired off of waivers from St. Louis on May 21.