Some characterizations of the Euclidean rectifying curves, i.e. the curves in $E^3$ which have a property that their position vector always lies in their rectifying plane, are given in [3]. In this paper, we characterize non-null and null rectifying curves, lying fully in the Minkowski 3-space $E^3_1$. Also, in considering a causal character of a curve we give some parametrizations of rectifying curves.