Rodway was born at the family home, Elsinore (named after the castle where Shakespeare's Hamlet is set), on Coliemore Road, Dalkey, Dublin, to Lillian Sybil (née Moyles) and Frank Rodway, who ran a shipping agency.

He made his first appearance in London's West End in 1959, as The Messenger in Cock-A-Doodle Dandy, and moved to England the following year.

His theatrical parts included Bassov in Summerfolk, and the title roles in Butley and Richard III.

With his expressive voice (described by Jack Adrian as "rich and dark and thumpingly Celtic"), he made many radio broadcasts for the BBC.

He played the role of Apemantus in both television and audiobook productions of Shakespeare's Timon of Athens.