Summary: The following generalization of distance magic graphs was introduced in [\textit{S. Cichacz} et al., Discuss. Math., Graph Theory 39, No. 2, 533--546 (2019; Zbl 1404.05185)]. A \textit{directed} \(\mathbb{Z}_n\)-\textit{distance magic labeling} of an oriented graph \(\overrightarrow{G}=(V,A)\) of order \(n\) is a bijection \(\overrightarrow{\ell}\colon V \rightarrow \mathbb{Z}_n\) with the property that there is a \(\mu \in \mathbb{Z}_n\) (called the \textit{magic constant}) such that \[w(x)= \sum_{y\in N_G^+(x)} \overrightarrow{\ell}(y) - \sum_{y\in N_G^-(x)} \overrightarrow{\ell}(y)= \mu \text{ for every } x \in V(G). \] If for a graph \(G\) there exists an orientation \(\overrightarrow{G}\) such that there is a directed \(\mathbb{Z}_n\)-distance magic labeling \(\overrightarrow{\ell}\) for \(\overrightarrow{G}\), we say that \(G\) is \textit{orientable} \(\mathbb{Z}_n\)-\textit{distance magic} and the directed \(Z_n\)-distance magic labeling \(\overrightarrow{\ell}\) we call an \textit{orientable} \(\mathbb{Z}_n\)-\textit{distance magic labeling}. In this paper, we find orientable \(\mathbb{Z}_n\)-distance magic labelings of the Cartesian product of cycles. In addition, we show that even-ordered hypercubes are orientable \(\mathbb{Z}_n\)-distance magic.