In this paper, we study the componentwise linearity of edge ideals of weighted oriented graphs. We show that if D is a weighted oriented graph whose edge ideal I(D) is componentwise linear, then the underlying simple graph G of D is co-chordal. We give combinatorial characterizations of componentwise linearity of I(D) if the vertices in V+ are sinks or |V+|≤1. Furthermore, if G belongs to certain chordal graphs or G is bipartite or the vertices in V+ are sinks or |V+|≤1, then we show the following equivalence for I(D): Vertex splittable ⟺ Linear quotient ⟺ Componentwise linear.