Grimm attended Virginia High School in Bristol, Virginia, where he played for the school's baseball team. He was projected as a third- or fourth-round pick out of high school in the 2007 Major League Baseball draft, but he fell to the 13th round where he was drafted by the Boston Red Sox. He fell in large part due to his commitment to the University of Georgia. He opted against pro baseball, enrolling at the University of Georgia where he played college baseball for the Georgia Bulldogs from 2008 to 2010. In his freshman year he pitched 31 innings with a 10.91 ERA. His sophomore year was considerably better as he pitched 78 innings with a 4.15 ERA. In his third and final year at Georgia, he posted a 5.49 ERA in 77 innings.
Grimm was drafted by the Texas Rangers in the fifth round of the 2010 Major League Baseball draft out of the University of Georgia.
Grimm made his MLB debut on June 16, 2012, against the Houston Astros. He won his debut, pitching six innings, striking out seven, walking none, and throwing 88 pitches for 56 strikes. Grimm was the first pitcher to have seven strikeouts and no walks in a debut since Stephen Strasburg's debut on June 8, 2010.
On May 2, 2013, Grimm was named American League Rookie of the Month for April. During April, Grimm held a 2–0 record with a 1.59 ERA in 3 starts, striking out 15 and holding opponents to a .239 batting average in 17.0 innings.
In the first half of the 2013 season, Grimm went into the All-Star Break with a 7–7 record and a 6.37 ERA. His final three Rangers' appearances saw him go 0-2 and pitch 11.1 innings, giving up 23 hits, 6 walks, and 15 runs.
The Rangers traded Grimm to the Chicago Cubs on July 22, 2013, with Mike Olt, C. J. Edwards, and Neil Ramirez, in exchange for Matt Garza. He was optioned to the Triple-A Iowa Cubs on July 23 and lost his first start for Iowa, giving up 7 earned runs in two innings including six straight hits to start the second inning. When major league rosters expanded in September, Grimm was recalled to the Cubs. Grimm finished the season in the Cubs' bullpen, working 9 innings and posting a 2.00 ERA.
Going into 2014 spring training, it was announced that Grimm would move to the Cubs' bullpen due to the considerable depth within the Cubs' starting rotation. At the end of spring training, Grimm made the Cubs' opening-day roster as a relief pitcher.
On August 29, 2014, Grimm became the 68th pitcher in Major League history to strike out four batters in a single inning.
Grimm finished the 2014 season with a 5–2 record, a 3.78 ERA and 70 strikeouts over 69 innings pitched, working primarily as a middle reliever.