Ive ridden on lots of dirt roads in Mexico, and some (most) have a good enough surface, some though I was really, really glad to have 2in wide tires that make it so much easier riding over soft, loose or very rocky surfaces.
As your bike has 35mm tires, it probably will be ok on dirt roads, but it could sometimes be a bit of a pain in the derriere.
Hey, by the way, what bike is it?
It is a Ghost Square Trekking 4.8 at least 4 or 5 years old