Hefetz, Mütze, and Schwartz conjectured that every connected undirected graph admits an antimagic orientation (Hefetz et al., 2010). In this paper we support the analogous question for distance magic labeling. Let Γ be an Abelian group of order n. A directed Γ-distance magic labeling of an oriented graph ⃗ G = (V, A) of order n is a bijection ⃗ l : V → Γ with the property that there is a magic constant µ ∈ Γ such that for every x ∈ V (G) w(x) = ∑ y∈N + (x) ⃗ l(y) − ∑ y∈N − (x) ⃗ l(y) = µ. In this paper we provide an infinite family of odd regular graphs possessing an orientable Z n-distance magic labeling. Our results refer to lexicographic product of graphs. We also present a family of odd regular graphs that are not orientable Z n-distance magic.