The road above the old Hotel Belvedere and Orsula Beach offers some good car-free running in Dubrovnik, with great views of the water. A good starting point is Jacob Church (Sveti Jakov), and head south on the un-named road until it runs into the D8 (highway). This part of the road is closed to traffic. It’s just shy of 2 km one-way.
For a longer run, continue north of Seveti Jakov, along Ul. Frana Supila, towards the Poce Gate (Vrata Od Ploča) to the old city. There are cars, but a wide enough sidewalk/shoulder that makes it safe for running. The total road between the D8 and the Gate is 3.5 km (2.2 miles), and is a gradual downhill heading north. Great views as you head toward the old city.