The shear problem for nematic polymers consists in characterizing all stable stationary orientational distributions, steady and unsteady, versus shear rate and material parameters. Continuum theory [cf. Leslie (1968), Ericksen (1960)] provides formulas for the shear response of liquid crystals in terms of a single viscosity ratio, the Leslie tumbling parameter lambda(L). Kuzuu and Doi (1983, 1984) developed a weak-flow asymptotic analysis of kinetic theory,,which gives a molecular basis for all continuum theory parameters. In this paper, we develop a mesoscopic extension of the Kuzuu-Doi method, applicable to any tensor model. Our method yields orientational and theological features of nematic polymers in weak shear with explicit formulas, parametrized by the parameters of the second-moment tensor model. This provides an explicit mesoscopic theory solution to the problem posed by Marrucci and Greco (1993) of how orientational degeneracy of quiescent nematic equilibria breaks in weak shear, leaving a finite set of steady stationary states, whose number, type (in-plane, out-of-plane), stability, phase transitions, and theological properties scale with parameters of the model. An intriguing feature to resolve is the multiple transitions associated with distinct steady distributions (logrolling, in-plane flow alignment, out-of-plane alignment), each with its analog of the Leslie criterion \lambda(L)\ = 1. We illustrate our method and its physical predictions by solving the weak shear problem for the Doi quadratic closure model, whose material parameters are nematic concentration and molecular aspect ratio. The predictions are confirmed with numerical simulations of the model, and compared with experimental data in weak shear from the review article of Burghardt (1998). We further predict scaling properties due to changes in concentration and aspect ratio that are less readily available from experiments. (C) 2003 The Society of Rheology.