Anthony Quinn (Alexis Zorba) had a broken foot during filming, and thus couldn't perform the dance on the beach as scripted, which called for much leaping around. The dance is called "syrtaki", and contains elements from various traditional Greek dances. It was created especially for this movie.
Simone Signoret was the original choice for Madame Hortense. After filming began, Director Michael Cacoyannis realized that she wasn't what he wanted for the part, and asked permission from Twentieth Century Fox head Darryl F. Zanuck to replace her. He agreed and he proposed Tallulah Bankhead, Bette Davis, and Barbara Stanwyck. Cacoyannis, though, had Lila Kedrova in mind. Zanuck had no idea who Kedrova was, or how she even looked, but he trusted Cacoyannis very much, so he agreed.
Such was the interest in an adaptation of Nikos Kazantzakis's hugely popular novel, this movie was already in the black before it opened.
In the earlier stages of filming, Director Michael Cacoyannis and Anthony Quinn had frequent disagreements, as Cacoyannis felt that Quinn was being too over-the-top.