Shear viscosities as well as first and second normal stress differences of solutions of rigid spherocylindrical colloids are investigated by Brownian dynamics simulations for aspect ratios L/D in a range from 25 to 60 and scaled volume fractions L/D from 0.5 to 4.5. Shear thinning behavior is observed in all cases. In the isotropic phase, the calculated viscosities at low volume fractions are in agreement with predictions by Dhont and Briels, while over a larger range of shear rates they are described by the Hess equation. The self-rotational diffusion coefficients obtained from the flow curves agree very well with those calculated by traditional methods. In the nematic phase, the inflection point of the flow curve is associated with the critical shear rate at which the orientational director changes its motion from kayaking to wagging. The first normal stress difference N1 in the nematic solution is positive at low and high shear rates but negative at moderate rates, which is rather distinct from the monotone behavior shown by isotropic solutions. The simulated second normal stress difference N2 is found much smaller in amplitude than N1 and always opposite in sign. Our findings qualitatively confirm existing theoretical predictions and experimental measurements. A newly developed event-driven Brownian dynamics algorithm, in which the excluded-volume interactions between particles are incorporated as collisions instead of as repulsive potentials, has made these simulations feasible.