Since your ancestors were Greek you are entitled to receive Greek citizenship. Because this citizenship is offered to you vis-a-vis applying for it, you do not loose your American citizenship.
Greek citizenship will give you access to a residence and work permits in any European Union country. This is extremely attractive to European employers because they do not have to use their quotas of non-European employees that they hire when they employ you. Another benefit is that they do not have to go through the three to six month waiting period that usually takes national authorities to approve a working visa/permit. Another country that offers its citizenship to descendants of its citizens like Greece is Ireland.
For more info. please visit: www.greekembassy.org
I hope that this helps.