While most neighbourhoods in Athens are safe particularly around the tourist areas, there are a few places that can be a little bit sketchy. Some neighbourhoods are only dangerous under particular circumstances, while others aren't really that great any time of day.
The most famous "bad neighbourhoods" of Athens are the areas around Omonia square and Exarchia. It's true that these neighbourhoods are a little sketchy, however during the day for the most part they’re fairly safe. Syntagma square gets a bad rap based totally on the fact that it's the square where people protest. If you're around the square and you see a protest, it's probably safest if you just walk away, however if nothing is going on, this is a great square to explore.
If you decide that you want to walk from the main tourist area towards the Archaeological Museum, you'll pass by several University buildings on 28 October street. It used to be that this area was a no go zone for police (due to an old law) and because of this it was a really sketchy area. Since the law was changed the police have cleaned up this area, and it's much better now. It's still a little sketchy, but during the day it's not as dangerous as it once was.
The one area that continues to keep its bad reputation is Agio Pandelimonas. This is probably the only neighbourhood that I would generally recommend avoiding. Not only is it fairly sketchy, but truly there really isn't much to see up there for the average tourist.