Greenaway began his career working for the Jim Henson's Creature Workshop on movies such as Dark Crystal (1982) and Labyrinth (1986). He was one of the main puppeteers, alongside Don Austen, David Alan Barclay and Sue Dacre among others, for Little Shop of Horrors (1986). He was also one of the assistant puppeteers on Yoda in Return of the Jedi.
In television, Greenaway worked on several Jim Henson Company productions, including The StoryTeller, and The StoryTeller: Greek Myths, the TV movie Living with Dinosaurs (assisting Brian Henson on the puppet dinosaur Dog), and the sitcom Dinosaurs (as facial puppeteer for many characters).
His other movie work included The Witches (based on the Roald Dahl book); Teenage Mutant Ninja Turtles, where he was the facial puppeteer for Raphael; and Babe. During a three month visit to Berlin in early summer 1995, he also played Mark Parill, a recurring character in eight episodes of the German TV drama Montagskinder .
In 1999, Greenaway was selected by Frank Oz to once again help him with Yoda's puppetry in The Phantom Menace. He handled the subtle movements on the character's face and body. That same year, Greenaway has performed as one of the creatures in John Paizs' science-fiction movie Top of the Food Chain.