I beg to differ here.. According to Wikipedia.. And i put in quotes..
"The Byzantine Greeks continued to call themselves Romans until their fall to Ottoman Turks in 1453. That year the Roman Empire was ultimately ended by the Fall of Constantinople. Constantine XI, emperor of the Byzantine Empire during 1453 is considered the last Roman emperor." And that is hardly in ancient times.
The "Roman" empire influenced many countries and for many years.
And then the Greeks might like belly dancing because it is fun? And it might make them a few extra dollars? It might have been ok for women of other cultures to dance in public?
And the Arabs influenced alot of the Mediterrean from Turkey to Spain. Including Greece and Egypt. The Rom are also another factor when it comes to music and dance.
I think the fact is that since there is no written or oral history of this form of dance we may never know why and how..

What a great thread!!

Thanks for bringing up this topic and while I really don't have much too add to Shira's [excellent] response to the question, I would like to say that my own [local] dance community might not have even existed had it not been for the Greek community here. It was the Greek restaurants and 'bouzoukias' [Greek nightclubs] that put bellydancing on the 'map' in our city and this specific ethnic 'scene' was the single greatest influence on its development here over the past 30 years. Greek people in general, don't necessarily think of bellydancing as any kind of 'high art form'...at least not in the way that the professional dancers themselves might. Their attitude toward it all is rather casual at best, although they do enjoy the spectacle of it [but it's more about the music for them]. Besides, many Greek women tend to feel that it just comes 'natural' them anyway, "so what is the big deal?"

I know from my own experience and long time involvement within the Greek restaurant scene [here], that bellydance shows had served a [very specific] purpose ....That of creating a diversion for diners [during that critical window of time] when the kitchen and server staff were swamped and needed time to catch up on orders. It was cheap, live entertainment [no hassles with unions or cabaret liscensing issues etc] AND..it was a nice little tax 'write off' too. This seemed to work very well for the establishments that hired dancers and it caught on.

Nowadays, this same restaurant dancing scene seems to have gone by the wayside and there aren't near as many venues featuring bellydancing as there used to be. However, I did notice while visiting Greece a couple of years back, that there is alot of Arabic/Greek 'fusion' in music and pop culture going on there and arabic "pop" music was featured in the nightclubs everywhere we went. Lots and LOTS of "bellydance" style clothing was being donned by young women and sold at all the department stores. Interesting eh? BTW...a good source to go to on this subject is Chryssanthi Sahar, a professional dancer of Greek descent and who is based in Germany. Here is her website: www.chryssanthi.com/