  Let $\mathcal L=-\Delta+V$ be a Schr\"odinger operator, where $\Delta$ is the
Laplacian on $\mathbb R^d$ and the nonnegative potential $V$ belongs to the
reverse H\"older class $RH_q$ for $q\geq d$. The Riesz transform associated
with the operator $\mathcal L=-\Delta+V$ is denoted by $\mathcal
R=\nabla{(-\Delta+V)}^{-1/2}$ and the dual Riesz transform is denoted by
$\mathcal R^{\ast}=(-\Delta+V)^{-1/2}\nabla$. In this paper, we first introduce
some kinds of weighted Morrey spaces related to certain nonnegative potentials
belonging to the reverse H\"older class $RH_q$ for $q\geq d$. Then we will
establish the boundedness properties of the operators $\mathcal R$ and its
adjoint $\mathcal R^{\ast}$ on these new spaces. Furthermore, weighted
strong-type estimate and weighted endpoint estimate for the corresponding
commutators $[b,\mathcal R]$ and $[b,\mathcal R^{\ast}]$ are also obtained. The
classes of weights, the classes of symbol functions as well as weighted Morrey
spaces discussed in this paper are larger than $A_p$, $\mathrm{BMO}(\mathbb
R^d)$ and $L^{p,\kappa}(w)$ corresponding to the classical Riesz transforms
($V\equiv0$).
