‘s’ is natural to me, but the ‘z’ doesn’t sound particularly jarring.
Usually Greece-y. But I’ve been known to use the other form for imagery purposes. Greezy just sounds nastier. And you can elongate that Z along with the E while still varying pitch.
My logic, if you care: greece is the noune, while greeze is the verb. The -y suffix is usually added to the noun, not the verb.
I was surprised and happy to see this thread as I heard that ad on the radio just this morning. Hearing ‘gree-zee’ was kind of jarring. I’m definitely a ‘grease-ee’ type of person.