Brownian dynamics simulations were employed to investigate the rheological properties and structure of suspensions of rigid spherocylinders, as a model for rodlike colloids. The spherocylinders interacted only through a soft repulsive force that mimicked a hard spherocylinder interaction. The translational and rotational diffusivities of hard spherocylinder suspensions were reproduced. Liquid crystalline phases, including isotropic, nematic, and smectic and solid phases were identified using orientational and hexatic order parameters and pair distribution functions. Typical flow curves observed experimentally for rodlike colloidal suspensions were reproduced in the simulations, with two shear thinning regions that bracketed a viscosity plateau at intermediate Peclet numbers. The transient rheology and structure of suspensions that were nematic at rest exhibited a variety of behaviors that depended on the Peclet number and concentration. Systemwide domains that aligned in and kayaked about the vorticity direction, domains that rotated coherently locally, and layered domains were observed. Oscillations in the order parameter, viscosity, and the first and second normal stress differences were associated with changes in the structure.