Since I´m not american, I can´t say if you as an american will feel safe in Greece, Italy and Turkey but I can say if you have any reasons not to feel safe.
Italy: I think that Italy have some small troops in Iraq too so I don´t think that they have anything particular against americans because of the war.
Greece: I haven´t heard about any big protests about the war there, I don´t think you have any reason to be scared there either.
Turkey: Maybe a problem, Turkey is a muslim country. Turkey wants to become a member of the european union so the goverment probably do what they can so that americans and people from northern and western europe shall feel safe there, I´m not sure what the people who lives there think.... The main problem for tourists, at least when it comes to safety, in Turkey are the Kurds, and they didn´t liked Saddam so they should like you!
Me myself don´t want to go to Turkey but that´s not for safety reasons, it´s more that I don´t like how the goverment there theats people who they disagree with (=Kurds)!