How do you cheer or say “Hooray!” in your language?

Greek.

Ζήτω! Zito!

Now, have I ever written a Quora post on how you say something in Greek, without a detailed disquisition on etymology and alternate expressions?

I won’t this time either.

Zito! is a third person imperative of zo, “to live”: so “may he live!” The third person imperative would certainly have died out by 1000 AD. And “may he live!” looks suspiciously like German Es lebe!

Yes. Like any number of other Modern Greek formulaic expressions, Zito! is a German expression, translated into Ancient Greek. Just like εντάξει endaxi = “OK” is the Ancient Greek translation of In Ordnung.

The Byzantine equivalent, which doesn’t have as much to do with German, is Εἰς πολλὰ ἔτη Is polla eti, “For many years!” That expression survived into modern Greek dialect—and Bulgarian—as Σπολλάτη! Spolati!