  We investigate connected cubic vertex-transitive graphs whose edge sets admit
a partition into a $2$-factor $\mathcal{C}$ and a $1$-factor that is invariant
under a vertex-transitive subgroup of the automorphism group of the graph and
where the quotient graph with respect to $\mathcal{C}$ is a cycle. There are
two essentially different types of such cubic graphs. In this paper we focus on
the examples of what we call the alternating type. We classify all such
examples admitting a vertex-transitive subgroup of the automorphism group of
the graph preserving the corresponding $2$-factor and also determine the ones
for which the $2$-factor is invariant under the full automorphism group of the
graph. In this way we introduce a new infinite family of cubic
vertex-transitive graphs that is a natural generalization of the well-known
generalized Petersen graphs as well as of the honeycomb toroidal graphs. The
family contains an infinite subfamily of arc-regular examples and an infinite
family of $2$-arc-regular examples.
