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.