What amuses me is all this talk about the so-called "proper" pronunciation of this word, and everybody has the last part of the word completely wrong.
In Greek, you don't create plurals with a final "s". The word is "gyros". As in, "I'll have a gyros sandwich". It should always end in "s". That includes the pronunciation, too. And it's not the "z" sound you make when saying English plurals. It's a real "s", like the first letter in "soft".
Of course, a reasonable person would point out that when words change language, they often get altered. No kidding. If this is correct, then this word should be gyro/gyros, and pronounced in the normal English way. "Gyroscope" doesn't have some silly, "correct Greek" pronunciation. And guess what? They're from the same Greek root (referencing the turning of both). But when "gyroscope" entered English, there wasn't a silly marketing campaign to try to confuse people. That would be the marketing campaign to make this dish sound like some authentic old Greek food. That's nonsense - this sandwich was invented in the US (either Chicago or New York).
It makes complete common sense to pronounce this word with correct ENGLISH pronunciation. That's the language we speak here. We don't say "Par-ee", we say "Paris" - and there's a million other examples of this. Struggling to try to say this according to some snobbish nonsense rules that no one can even agree to (while Anglicizing the ending of the word completely anyway) is beyond foolish.