An alternative computational adaptive method to solve the Leslie-Ericksen equations of nematic hydrodynamics is presented. The method uses adaptive torque balances and is able to accurately compute arbitrary three-dimensional orientation fields. The method is applied, in conjunction with computational bifurcation methods, to solve the governing equations for a model rigid-rod, nonaligning, nematic polymer, in steady and transient rectilinear simple shear flows, using fixed parallel director anchoring. The five-component solution vector consists of the primary and secondary velocity components and the three-dimensional director field n. The parameter space is the line representing the magnitudes of the Ericksen number (scrE). According to the magnitude of scrE, seven types of stable steady-state solutions are found and fully characterized. The seven types of solutions are classified as in-plane solutions if the director remains within the shear plane, defined by the flow direction and the velocity gradient, and as out-of-plane (OP) solutions if the director field is out of the shear plane (three-dimensional orientation). The six OP solutions are three pairs of mirror-image solutions that differ from each other by their rotation number (\ensuremath{\Lambda}). Two pairs of out-of-plane solution branches are achiral (\ensuremath{\Lambda}=0) and display one-way twisting from the shear plane. One pair of out-of-shear-plane solution branches is chiral (\ensuremath{\Lambda}=\ifmmode\pm\else\textpm\fi{}1) and displays a full 2\ensuremath{\pi} director rotation when going from the bottom plate to the top plate.The nucleation mechanisms of these chiral branches are identified using a torque analysis. The main bifurcation phenomena and the local stability of the branches are summarized in a bifurcation diagram. The main structural changes, as the parameter scrE increases, are captured by visualization of the transient director field. The main features of the velocity field are captured by particle tracking visualization, which yields three-dimensional particle motions driven by the combined primary and secondary flows. The main mechanical responses, captured by the transient rheological functions (apparent viscosity and the first normal stress differences), exhibit, at low shear rates, non-Newtonian responses that are usually found in isotropic, viscous, or viscoelastic liquids, at large shear rates. Dynamic simulation in conjunction with the bifurcation diagram is used to construct the stability phase diagram, which yields the adopted stable steady-state solution that results from shearing a monodomain sample with fixed parallel director anchoring. Finally, the strain history is shown to control the finally adopted steady state, thus invalidating linear superposition which is usually valid at small shear rates.