The Green Coast, or Costa Verde, is considered one of the most amazing places of the island. Until the first half of the XX century was an important mining center (also my grandad used to work there), but once this activity wasn't that profitable anymore, mines closed, people left and nature flourished.
Green coast takes its name from the impressive vegetation, and to get there you first have to pass some mountains. On the way to Costa Verde you'll find many abandoned mining ghost towns, by this time desert, that makes the enviroment more similar to a western movie. But once you start going downhill, the view on the sea is breath-taking.
Costa Verde is perfect if you want to stay in the nature and enjoy of the silence. Imagine than in this part of Sardinia, exactly in Piscinas, you can find the biggest dune sand of Europe.
As the local economy based on mining failed, people start to develop a low-scale form of sustainable tourism, far from the fancy-posh-disneyworld style of the tourist resorts in the north-eastern part of Sardinia.
Small guesthouses and bed and breakfast run by locals, offering local products, make the stay in this part even more unique.
I guess it's even possible to rent a small flat in some super tiny places near the sea.