  A virtual link diagram is called mod $m$ almost classical if it admits an
Alexander numbering valued in integers modulo $m$, and a virtual link is called
mod $m$ almost classical if it has a mod $m$ almost classical diagram as a
representative. In this paper, we introduce a method of constructing a mod $m$
almost classical virtual link diagram from a given virtual link diagram, which
we call an $m$-fold cyclic covering diagram. The main result is that $m$-fold
cyclic covering diagrams obtained from two equivalent virtual link diagrams are
equivalent. Thus we have a well-defined map from the set of virtual links to
the set of mod $m$ almost classical virtual links. Some applications are also
given.
