How do Greeks say “Happy New Year”? Is there more than one way to say it?

Yup:

• Καλή Χρονιά, “Happy [New] Year” (the most frequent form)
• Καλή Πρωτοχρονιά, “Happy New Year’s Day!” (specific to the day)
• Ευτυχισμένο το Νέο Έτος, “May the New Year be Happy” (a more official formulation, of the kind you’ll see in writing or address to dignitaries)