Although the clear blue water and whitewashed buildings give one a sense of tranquility, there are dangers to be avoided in certain locations and certain times of year. Greece is a very safe country for tourists, but it's best to be aware of the hotspots that exist for both travelers and locals.
Places that might prove to be unsafe for American travelers include football (soccer) games where the rivalry between fans of the two teams is at a peak. Greeks take their football very seriously. It's not uncommon for the fans of the losing team to pull up stadium seats, throw them onto the playing field, set fires, and in general, behave worse than a 3 year old who has been denied a toy in the supermarket. Innocent spectators have been killed and injured in this sort of riotous behavior at games and in the vicinity of the stadium.