Athens is generally a very safe city and its crime rate is lower than other capitals. Nevertheless, here are some things you should beware of:  

Pickpockets

As in every capital, there are pickpockets who are preying especially on tourists. They are very swift, often operate in teams, and are particularly present in the metro, in the train from the airport, and around Monastiraki Square. So use caution generally, watch your camera and your mobile phone and always carry your money and identifying papers in a inner belt (NOT in a side pocket, even zipped). Be extra vigilant in crowded places, on escalators, when taking out money from ATM's or buying metro tickets from ATM machines. A known trick is that one will push you, ask a question, or otherwise distract you, while the other will "visit" your pockets and pass it quickly to a third fellow.

Drivers in Athens

As you will quickly realize, driving in Athens can be chaotic. Cars, motorcycles and pedestrians  swarm the streets and pretty much disregard traffic laws, traffic lights, etc. Don't be surprised by seeing cars continuing after the traffic light has become red, motorcylces driving in the wrong direction of a one-way street, pedestrians crossing in middle of traffic, cars parked pretty much everywhere...
As you are walking in Athens, keep this in mind, and make sure cars are fully stopped before crossing a street. Drivers also don't necessarily yield for pedestrians when turning into a street... So in a few words, remain calm, be careful and expect the unexpected!