They fly to Larnaca as well don't they? I should think so, there is definitely a market to exploit in the Middle East, bearing in mind the number of American expatriates, government and military personnel working in the area.
Saudi Arabian, Egyptair, Kuwait and soon Emirates will be/are serving the US market. Interestingly, not all American citizens fly home with a US flag carrier. I often meet Americans on the BA flight from Riyadh.
PS i've got a lovely view of the lunar eclipse here-it's going bright red!