Greene followed a curious path to the acting profession. During his high-school years, he says, he never thought about being an actor. Instead, he wanted to be a draughtsman--and he actually did that for a time in his youth, along with other construction-related and manual labour jobs. Eventually he found himself working as a tech guy in the music industry, touring around with bands. It was only when an acquaintance harassed him incessantly to appear in a play he'd written--Greene even once turned out the lights to pretend he wasn't in when the man showed up at his office one day--that he got his first acting role.
Years of working in small-scale stage productions in Canada and Britain followed, until, one break leading to the next, Greene eventually found himself nominated for an Oscar for his work as Kicking Bird in Kevin Costner's epic Dances With Wolves.