The guy that played the most had to be Anthony Quinn.
Off the top of my head I can name about eight that the Mexican/American actor portrayed.
Arab – Lawrence of Arabia
Mexican - Viva Zapata
Greek - Zorba the Greek
Italian - The Salamander
Native American - They Died with Their Boots on
Spanish - Seven Cities of Gold
North African - Road to Morocco
Filippino - Back to Bataan
French - Lust for Life
American - The Happening (the first one)
(Yes, I know he doubled or tripled with many of these)
I also remember him playing at least two Asians, a Polynesian and I think a couple of South Americans but I am not sure.
I imagine there were a number of other actors who qualify with at least five, but who are they?