Light can change the orientation of liquid-crystal molecules. Usually, the torque that causes the reorientation originates in angular-momentum transfer from the radiation field to the material. If a small amount of dichroic dye is dissolved in the liquid crystal, a light-induced torque can appear essentially without the transfer of angular momentum from light. We show that, in such cases, the dye molecules act as light-driven molecular motors which, via an orientational Brownian ratchet mechanism, transfer angular momentum, which originates at the cell walls, to the liquid crystal. Understanding the details of this mechanism is important for applications ranging from flat-panel displays to optomechanical transducers.