The 30-year-old Rodriguez had 18 strikeouts in 16 innings for Nashville this season, compiling a 2.81 ERA and holding batters to a .143 batting average.
Athletics manager Bob Melvin has struggled to find the right roles and combinations for his relievers. The group has a 5.32 ERA with five blown saves in nine chances this season. Tyler Clippard, who is 3 for 4 on saves, and right-hander Evan Scribner have been their only consistent relievers.
To make room for Rodriguez, the A's designated right-hander Chad Smith for assignment. Smith's fate was sealed when he gave up four runs in 1-3 of an inning during the 13-0 loss to the Minnesota Twins on Wednesday night.