I travel around the globe reasonably often and I read and consider the warnings but I let the news and an internet search be my ultimate guide. I do not take unnecessary chances such as going to Egypt this weekend but I would not be that concerned in Istanbul even with the recent warnings. (edit - Istanbul is not on the list now)
I was in Athens during some of the riots last year, most of the pictures you seen were right around the hotel I stayed in. I of course stayed out of them as getting anywhere near the riots is begging for trouble but the next morning, I had my breakfast with the locals less than a block away and was warmly welcomed. The problem was with the government, not me and of course the money we send there is welcomed.
Should I have been in some muslim shit storm in the middle east - I am sure I would not have been safe. Let the location/situation be your guide.