I don't know about biking in Rome other than urban biking, nor do I know its outskirts. Umbria, however, is similar to Tuscany (with fewer international tourists). There are charming hilltop towns everywhere. You could try Gubbio, Todi, Orvieto and CivitÓ di Bagnoregio outside of Orvieto. Assisi can be overrun with tourists, including Italian religious tourists, but otherwise is also a hilltop town you can ride up to from the surrounding countryside as well as down from.
Todi has an easy winding way up and a hard, unyielding straight climb into the center of the old town. CivitÓ is a virtual ghost town that you ride up this ramp-in-the-air across a gorge to get into the old town—very atmospheric. Orvieto is perched on top of a volcanic "tufa" cliff, has its cathedral and network of Etruscan tunnels under it.
On the whole, if you avoid the SS roads ("statale" in Italian), the regional SR and provincial SP roads tend to be pretty quiet. The SS roads can be either quiet country roads with light traffic or major arteries with heavy truck and motor traffic.