  In this paper, we extend work of the first author on a crystal structure on
rigged configurations of simply-laced type to all non-exceptional affine types
using the technology of virtual rigged configurations and crystals. Under the
bijection between rigged configurations and tensor products of
Kirillov-Reshetikhin crystals specialized to a single tensor factor, we obtain
a new tableaux model for Kirillov-Reshetikhin crystals. This is related to the
model in terms of Kashiwara-Nakashima tableaux via a filling map, generalizing
the recently discovered filling map in type $D_n^{(1)}$.
