[1] Born in Maryhill, Glasgow, Grieve attended the Royal Scottish Academy of Music and Drama, before joining the Citizens Theatre in 1951.

[1][2] Grieve worked in variety alongside many familiar Scottish comedians, including Stanley Baxter and Jimmy Logan.

[3] Although principally known for his comic roles, he appeared in drama films such as The Thirty-Nine Steps (1978), Eye of the Needle (1981) and the BBC docudrama Square Mile of Murder (1980).

[6] He played Sandy Duncanson in BBC's adaptation of Neil Munro's The New Road, in a BBC drama about the Union of the Parliaments in 1707 he played John Hamilton, 2nd Lord Belhaven and Stenton who delivered a controversial speech against the Union, and appeared on BBC Scotland's Hogmanay celebrations,[7][8] one of which (Into '85) was broadcast nationally from Gleneagles and became notorious for Grieve, apparently worse the wear with alcohol, unable to recite a brief poem and collapsing into laughter, along with other shambolic incidents featured in the same programme.

[9][10] He appeared in two episodes (eleven years apart) in the television series All Creatures Great and Small as Dr. Harry Allinson,[11] whose practice was next door to Skeldale House.