You'll naturally come to this area in the afternoon or early evening, when you'll probably be strolling the pedestrian promenade by the Acropolis. From there it's a short detour up Filopappou Hill to admire the Acropolis.
But if you make a special trip in the morning, you may find the wonderful Byzantine Church of Agio Dimitrios Loumbardiaris open, and you'll have a little more light and time to appreciate the beautifully paved paths. The adjacent Hill of the Pnyx, very little visited, is also a peaceful place any time of day, with lots of bird life and equally good vantage on the Acropolis.
Thisio and Petralona are both pleasant, distinctly un-scene-y places for lunch or dinner, and especially rewarding for anyone who prefers wandering a bit to get a feel of what's available. Petralona has an upper ('Ano') and a lower ('Kato') portion; Iroon is the main street through the upper part.