Born in Allans Flat, Victoria, he was educated at Williamstown High School, then matriculated to the University of Melbourne.
He remained there until 1947, except for a period in 1938 and 1939 when he studied mycology at the University of Cambridge, and a brief time serving in the Royal Australian Naval Reserve early in World War II.
During World War II his university research included an investigation into fungal contamination of field glasses in New Guinea.
His research interests were broad, taking in general botany, anatomy, physiology, genetics, biosystematics, ecology, mycology and systematics.
Grieve was a long-time member of the Royal Society of Western Australia, joining in 1948, and twice serving as President.