The Plaka is the oldest city in Athens and is right in the "shade" of the Acropolis. Mostly for tourists, apparently it still has a few gems of artisan style shops. Great place to wander for postcards.
You won't get major exercise from this but you'll sure get some great scenery. Most people bring picnics along and stop by the side of the river. Sounds like something out of a Merchant- Ivory movie, doesn't it?