I live in Athens and I can tell you that it's pretty safe here. There are a few neighborhoods I would avoid in the evenings (Omonia and around the universities), but the tourist areas are fine. The only thing you should do is pay attention to protests in the news. Most of them happen in the Omonia area and around Syntagma square. If you hear of a protest (ask at your hotel) then just avoid those areas. Places like the Acropolis, and the Plaka are still safe when there's a protest.

The only other inconvenience you might encounter is the strikes. Again, ask at your hotel, and plan accordingly. Make sure you don't plan on heading out far if there's a metro strike because it might be difficult or expensive to get back.

The two best resources for information are:
http://www.ekathimerini.com/
http://livingingreece.gr/strikes/