Source: http://www.bellydancingvideo.com/greek-belly-dance.htm

Oftentimes, the Greek singer will feature an amané section or song, where he or she will improvise in a very Middle Eastern manner, repeating the word "Aman" which literally means "have mercy!" The amané is very similar to the Turkish gazel. It is the Turkish influence that gives its Middle Eastern influence to the Greeks' tsiftételi.

The Balkan countries were long under Turkish rule, and they brought along their music and culture, which was absorbed by the countries they occupied.

Because Greeks and Turks have not always gotten along throughout their long history, the Greek government actually censored music in the 1930's that featured Turkish and other Middle Eastern elements, such as the amané and rembetica, an underground music that was popular in the hashish dens of port cities. Even the bouzouki was looked down upon in those days. Early versions of the popular Greek bouzouki had moveable frets and were tuned with drone strings, which allowed the musicians to play exotic sounds similar to that of the Turkish saz and Lebanese buzuk.


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Note added at 12 mins (2010-09-27 17:56:39 GMT)
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If "amané" needs to be described in your translation, then I would describe it as Turkish-themed belly dancing music.