I am not sure why the area around Syntagma should be avoided other than the fact that's it's a main metro station and it's where parliament is so there are often demonstration there. On the other hand, it's right in the middle of everything.
Anyway, I'd probably avoid the area around Omonia but other than that, you'll be just fine.
Again, greeks are lovely and helpful people. You'll be fine.
Just remembered, one time we stayed at a really lovely hotel right near Syntagma called Amalia Hotel Athens, I highly recommend it.