I ran 2 Greek campaigns (still European, but you know) in 3.5. It went okay, but ran into a weird issue with monsters. For some reason, it's fine for a sphinx and a chimera to show up in your mostly-northern-european campaign, but when you focus on Greece your list gets small and takes a lot of work. Trolls seem out of place and dragons aren't major players, for example. This was my early DMing days, but might be a hurdle for new GMs taking on a different or focused setting.
I've done modern games (Shadowrun, etc) across the globe, though that's a bit easier, as basically modern cities are all the same.
I'm currently running an Arabian Nights themed campaign. My players are pulling almost entirely from Aladdin and Prince of Persia and I spend a short time every session on some element of culture. I.E., the first puzzle they encountered required name ordering using "ibn" and "abu".
It feels less like exposition and more learning by doing that way.
Bring in elements that interest players specifically, as with any campaign. The alternate cultural settings should allow for different flavor, themes, and RP. Slavery is a strong theme in this campaign, as genies and their bottles are the central focus. But each player, unwittedly or not, has given me their own form of slavery - responsibility to a family/tribe, born into a position that won't inherit, and stuck in a job they hate. These things aren't exclusive to any setting, but can be focused and stylized in a different way thanks to the setting here.
Rambling, I know, but I guess my point is to find a way to emphasize themes from that era's mythology, and sprinkle in real facts or setting details along the way.