Grissom attended Paul J. Hagerty High School in Oviedo, Florida, where he and Riley Greene were teammates on the baseball team.

[5] Grissom made his professional debut with the Rookie-level Gulf Coast League Braves, batting .288 with three home runs and 23 RBIs over 44 games.

[18] He made his MLB debut that night as Atlanta's starting second baseman at Fenway Park versus the Boston Red Sox, going 2-for-4 with a two-run home run over the Green Monster off of Darwinzon Hernández and a stolen base.

[17] Following the departure of Dansby Swanson in free agency, Grissom entered the 2023 regular season with a opportunity to become the Braves' starting shortstop.

[23] On December 30, 2023, the Braves traded Grissom to the Boston Red Sox in exchange for Chris Sale and cash considerations.