Canney was born in Falmouth, Cornwall. From 1940 to 1942 he studied at Redruth and Penzance and at St Ives School of Painting. From 1947 to 1951 he studied at Goldsmiths College School of Art, London, where he met Victor Pasmore. Later, on moving to the West Country, he became close friends with Patrick Heron, Roger Hilton and Peter Lanyon, also collaborating with Barbara Hepworth on a sculpture exhibition in 1957. By 1958 his work was influenced by abstract expressionism.
 
Canney lectured and wrote on contemporary art and he accepted a post as Lecturer at the University of California, in Santa Barbara. In 1966 he was invited by Paul Feiler to join the staff at the West of England College of Art, where he developed his constructivist ideas. From 1970 on he began to work in the Constructivist tradition using asymmetrical constructions and abstract forms. His white reliefs of the 1970s and 1980s were pure minimalism. Solo shows included Newlyn Art Gallery in 1983, Prescote Art and Design in Edinburgh in 1984, and The Belgrave Gallery, London, in 1990. An exhibition of later works was held at the Fine Art Society in 2007. His work is held in a number of public collections.