Popular depictions of ancient Rome and Greece show games of chance (dice-throwing, sports betting) and physical sports like wrestling and footraces and such. But was there any game of skill or strategy that occupied the cultural place of chess or Go or even Settlers of Catan?
Mercenaries (Rome), Senet (Egypt), Petteia (Greek, mentioned as a good thing to play for citizens by Aristotle) and Mancala (Egypt again). Senet and Mehen may be a bit old for your given time window, and Petteia and Mercenaries could well be two different versions of the same game.
Oh, and don't think that rules for these stayed the same - even chess rules changed massively since medieval times, the only thing that remained the same is the pieces, but their initial positions and movement patters are both different.