You don't have to be a great strikeout pitcher to strike out four batters in one inning. In fact, it helps if you're a bit wild, because in order to do so, you have to have at least one batter reach base on a wild pitch, passed ball or error.
October 14, 1908, Orval Overall: If this date sounds familiar, it's because that's the game in which the Cubs won that year’s World Series. Overall, who finished third in the National League that year with 167 strikeouts, threw a three-hit shutout to win the Series for the Cubs and struck out 10. Most games that long ago don't have play-by-play data available, but since it's a World Series game, we do know that Overall did it in the first inning, a wild inning in which he walked the leadoff hitter and allowed a single before Claude Rossman reached on a strikeout/wild pitch to load the bases. It was the only real threat the Tigers had in that game. At the time, it was just the fourth four-strikeout inning in big-league history and it still stands as the only one in the postseason.
May 27, 1956, first game, Jim Davis: From one of the best pitchers in Cubs history to "Who's he?", Davis had come into the game in relief in the fifth inning of a game tied 8-8. He immediately coughed up the lead run to the Cardinals on a Stan Musial RBI single. In the next inning, he struck out four. The runner reaching base was Cardinals pitcher Lindy McDaniel, and the play was ruled an error on catcher Hobie Landrith, since a run scored. It was the first time anyone had done this since 1916.
July 31, 1974, first game, Bill Bonham: A hard-throwing righthander who was sixth in the N.L. in strikeouts in 1974 — and fifth in walks — Bonham struck out four Montreal Expos in the second inning of this game. The batter who reached was, again, a pitcher: Montreal's Mike Torrez, who proceeded to steal second base. Bonham K'd the side to end the threat, but the Cubs lost 7-4. Bonham, who went 11-22 that year in a time when pitcher W-L records still meant something, is the last Cubs pitcher to lose 20 or more in a season.
September 2, 2002, second game, Kerry Wood: Must be something about doubleheaders, as three of these occurrences happened in one game of a twin bill. Wood's is unique among the Cubs' six because he allowed two runners to reach base on strikeouts, one on a throwing error by Todd Hundley, the other on a wild pitch. Thus Wood might have had a chance to strike out five batters in an inning, but the other out was recorded on a groundout. The Cubs won the game 17-4; in addition to the K mark, Wood hit one of five Cubs home runs in that game.
October 4, 2009, Ryan Dempster: In the season's final game when players could have been forgiven for having something other than baseball on their minds, Dempster struck out four Diamondbacks in the fifth inning. Gerardo Parra reached on a wild pitch and Dempster also issued a walk in that inning, but got out of it unscored upon. The Cubs lost the game 5-2; the only other notable thing that happened for the Cubs that day was Sam Fuld's first big-league home run, the only one he hit as a Cub. It's one of two dates on which two different pitchers did this: The Padres' Luke Gregerson did it against the Giants.
September 20, 2012, Jason Berken: Take your pick. Who's the most unlikely name on this list, Berken or Davis? I'd vote Berken, as he pitched in just six games for the Cubs and this was the only one in which he wasn't awful. In fact, he threw six shutout innings and for that, won my “unsung hero” award for 2012.
After his great outing, Berken got to watch Manny Corpas blow the game in the seventh inning by allowing five runs. Berken's 4-K inning allowed just the one baserunner, Ryan Hanigan, who reached on a wild pitch. Berken didn't have much of a big-league career, but he'll always have this little corner of the record book to show his grandchildren. And as in the previous Cubs occurrence above, another pitcher also K'd four batters in an inning on this date: Phil Hughes of the Yankees did it against the Blue Jays.
August 29, 2014, Justin Grimm: In a game better remembered for a pair of monster home runs hit by Jorge Soler, Grimm finished up the ninth inning by striking out four Cardinals. Future Cub Daniel Descalso was the extra-K guy; he reached on a wild pitch after Grimm had allowed a single to another future Cub, Jon Jay. Grimm then struck out Matt Carpenter and Kolten Wong to end the game. Grimm's 4-K inning marks one of just nine times a pitcher accomplished this feat in the ninth inning or later.
Here is Grimm’s 4-K inning: