I go to Corsica a few times a year, which is very close. Basically you will find lots of double track all over the place. But stuff we would call singletrack is harder to find – the population is smaller so paths are used less. Also the undergrowth / rocks make a lot of it hike a bike.
It is pretty much open access, but its not like here in that respect. They do a lot of hunting for wild boar, so take that into consideration on the weekends – or you may get shot 🙂
However, there will be some, its finding it that is harder.
While in Corsica last December I bumped into Naomi who guides for Saddle Skedadle and they do MTB guiding and Roadie stuff in Sardinia.
Try speaking to them. She told me about some of the old mining tracks they are having a lot of fun on with their mountain bikes.
Oh and June will be bloody hot, almost too hot unless its early in the morning / late afternoon, or high up in the hills.