Thyagaraya Nagar, known locally as T. Nagar, is a busy shopping district a couple of km west of the beach area. One can put together a decent run in this area, passing by and around three decent parks, linked by Gopathi Narayanaswami (GN) Chetty Rd., where there’s a collection of art deco bungalows.
A good place to start is at the Mambalam train stop. Head toward Natesan Park, which isn’t big but has a nice green space to run around. Then head slightly north to Panagal Park, which at 3.2-hectares features shaded paths and some fountains. Next along GN Chetty Rd. is Jeeva Park, which also has some nice walking trails. Further along is Semmozhi Poonga Botanical Gardens, which you can run/walk around. Nearby is a green area and some quieter roads around the U.S. Consulate. If you want to be ambitious, take Uthamar Gandhi Rd. north for 1 km to Haddows Road Park, which has a nice jogging path.