There are numerous unpaved county roads all over the hill country. The ones I have ridden on the most are in between Bandera and Utopia, but there are others that criss-cross up to Fredricksburg. Some cross private property lines, so do not be surprised to see cattle and other large critters wandering around.
The creek crossings can be difficult if it has rained recently, but otherwise, pretty easy going. Make sure you have your tank topped off in case you get a little lost. A GPS would help a bit, too.
IMO, the best way to find them is just to explore ... cruise down the roads you are normally going and look for any side roads ... explore.
I have been looking at some google maps of the area south of Rocksprings and there appear to be a large number of unpaved roads in that area, but I am not sure if they are open to the public or not. Would be a nice adventure to find out.