Running in the centre of Athens seems a bit daunting, truth be told, but there are a number of other options that are slightly outside the centre of town.
However I would like to let you know about one of my favourite running spots in the center of the city.
One of the best kept secrets of Athens running is the route above the Panathenaic Stadium (Kallimarmaro). The classic marble landmark is a stop on many tourist visits, but few know that a path runs along the top of the stadium (following the horseshoe shape) and there is also a park on the neighbouring hill with incredible views of Athens. If you are staying near the area of Pagrati or if you just like to run while you are in the city this is definitely worth a visit. The path itself is accessible from behind the stadium, through a gate on Archimidous Street, and its length is 500 meters. The surface is quite flat. As far as I know this is the only area in central Athens that is basically a track and a trail path.
Why do I like this place so much? Easy answer: there are always other runners for company, a lot of cats around and you have great view of the Acropolis, Likavittos and the mountains that surround Athens! The inspiration from the Stadium is a plus!