Qualifying as a primary school teacher in 1948, Greenwood began his career as a school teacher, but later moved on to Melbourne and Burwood Teachers' Colleges as a teacher of Art.
In 1968 Greenwood resigned from academia to pursue full-time writing and illustrating. Greenwood's picture books and novels reveal ideas on the themes of conservation and the pattern of existence. According to the Oxford Companion to Australian Children's Literature (1993), while Greenwood's illustrations are 'sometimes not immediately appealing, they are always very demanding. His use of colour is restrained. In both his pictures and his stories, he stretched the imagination and intellect of his readers'.
Apart from works published in his own right, Greenwood is also well known for his collaboration with Paul Jennings and Terry Denton on Spooner or Later, Freeze a Crowd and Duck for Cover. In addition to his literary pursuits, Greenwood worked part-time in horticulture, was the Craft reviewer for the Melbourne newspaper The Age for four years, and has also written for radio.