Another route that might fit your criteria is the Traunseetram, an 18.6 km line connecting the towns of Gmunden and Vorchdorf in Austria.
The equipment used would probably qualify as a tram rather than a train (although I'm by no means an expert), its tracks run on the street for the section in Gmunden, and along a road for the last few kilometres into Vorchdorf. It also offers a nice view of the mountains and the entire route most certainly qualifies as rural: Image source
For the most part, however, it does not run along a road like you specified.
You can find more information on the operator's website (German only), the (very short and uninformative) English Wikipedia entry, or its more extensive German versionGerman version.