  We consider cyclic $m$-isometries on a complex separable Hilbert space. Such
operators are characterized in terms of shifts on abstract spaces of weighted
Dirichlet type. Our results resemble those of Agler and Stankus, but our model
spaces are described in terms of Dirichlet integrals rather than analytic
Dirichlet operators. The chosen point of view allows us to construct a variety
of examples. An interesting feature among all of these is that the
corresponding model spaces are contained in a certain subspace of the Hardy
space $H^2$, depending only on the order of the corresponding operator. We also
demonstrate how our framework allows for the construction of unbounded
$m$-isometries.
