The dynamics and rheology of a rigid polymer or nanorod suspended in a Newtonian, viscous fluid under torsional flow have been studied. Our theoretical analysis predicts that rigid rods migrate in the radial direction according to their orientational coil figuration which is controlled by the competition between the shear flow, which tends to align the rods in the direction of flow, and Brownian motion, which tends to randomize the orientation. Steady and transient distributions of the center-of-mass in it dilute solution are derived from a kinetic theory and are confirmed by performing a Brownian dynamics simulation. The migration shifts the distribution toward the axis of rotation and enhances the shear-thinning behavior of the suspension.