A pioneering figure in hard bop, Griffin recorded prolifically as a bandleader in addition to stints with pianist Thelonious Monk, drummer Art Blakey, in partnership with fellow tenor Eddie "Lockjaw" Davis and as a member of the Kenny Clarke/Francy Boland Big Band after he moved to Europe in the 1960s.

[3] Griffin studied music at DuSable High School in Chicago under Walter Dyett,[1] starting out on clarinet before moving on to oboe and then alto saxophone.

[5] By mid-1947, Griffin and fellow Hampton band member Joe Morris,[1] had formed a sextet made up of local musicians, including George Freeman,[4] where he remained for the next two years.

[1] During this period, he recorded a set with Clark Terry on Serenade to a Bus Seat, featuring the rhythm trio of Wynton Kelly, Paul Chambers, and Philly Joe Jones.

His relocation was the result of several factors, including income tax problems, a failing marriage and feeling "embittered by the critical acceptance of free jazz" in the United States, as journalist Ben Ratliff wrote.

[15] In the late 1970s, Griffin recorded with Peter Herbolzheimer and His Big Band, which also included, among others, Nat Adderley, Derek Watkins, Art Farmer, Slide Hampton, Jiggs Whigham, Herb Geller, Wilton Gaynair, Stan Getz, Gerry Mulligan, Rita Reys, Jean "Toots" Thielemans, Niels-Henning Ørsted Pedersen, Grady Tate, and Quincy Jones as arranger.

[8][17] With Ahmed Abdul-Malik With Nat Adderley With Art Blakey With the Kenny Clarke/Francy Boland Big Band With Dizzy Gillespie With Philly Joe Jones With Thelonious Monk With Bud Powell With A. K. Salim With others