  We characterize all ruled translating solitons in Minkowski 3-space. In
contrast to the Euclidean space, we find ruled translating solitons that are
not cylindrical. These surfaces appear when the vector field that defines the
rulings, viewed as a curve, is a lightlike straight line. We also classify all
cylindrical translating solitons, obtaining surfaces that can be considered as
analogous to the grim reapers of Euclidean space, but also other surfaces which
have no a counterpart in the Euclidean space.
