Running along dirt paths
We stayed in Goreme, the most central town in the Goreme National Park. There are a number of dirt paths between the towns of Goreme and Cavusin. Our hotel was relatively close to Muze Cd., which runs east out of town. As you pass Cappadocia Adventures on the right, there will be a road to your left (i.e. going north). Turn on that dirt road and you can go a number of miles north passing the Sword Valley and going into Pasabag Valley. If you run in the morning and the balloons are running that day (all balloons are canceled if there is inclement weather or high wind), you will see dozens of balloons setting up or floating away along this route. There are a number of forks as you work your way north and you definitely want to keep track of where you are going on Google Maps or a trail app. There are also some ups and downs so I was able to do some hill work. I did get lost once and had to backtrack. Also, although most dogs are stray but friendly in Turkey, on this route I did once encounter a dog that barked at me (not friendly barking) and followed me for about half a mile. If you follow any trail north far enough (I usually went past Pasabag Valley), you will eventually come out somewhere along the road between Goreme and Cavusin, called Goreme Yolu
Road running
The roads between Goreme, Cavusin to the north and Uchisar to the south have stone sidewalks, so running the roads isn't dangerous. It's less scenic than the trail run I describe above, but you can't get lost and these roads are pretty flat. The road from Goreme to Cavusin (called Goreme Yolu) is about 4 kilometers and the road from Goreme to Uchisar (called either Goreme-Uchisar Yolu or Adnan Menderes Cd.) is just under 4 kilometers, so you can get in some miles by making a loop from any of these three towns and returning to that town. You can also take the Goreme Yolu north past Cavusin to Avanos, which is 4 kilometers past Cavusin. Finally if you want some hills, Uchisar is basically a hill town, so if you keep running into Uchisar you have to run uphill across various switchbacks. Most roads seem to lead up to Uchisar Castle, the high point in the Goreme National Park.