The nonequilibrium molecular dynamics computer simulation method was used to study microsegregated block copolymer systems in a selective solvent under a shear flow field. Two polymer concentrations were considered, 0.3 and 0.4, corresponding to the body centered cubic spherical and hexagonal cylindrical zero-shear phases, respectively. As the shear rate increased, both systems exhibited two-stage shear thinning, a peak in the scalar pressure, and normal stress differences. Microscopic connections were investigated by calculating the gyration and bond orientation tensors and the interaction energies per particle. At high shear rates, polymer chains elongate and orient along the direction of shear, and this is accompanied by the breaking-up of domains. The structure-rheology relation was discussed with regard to the morphological changes reported in our last study for the same systems. In particular, the structurally relevant critical values of the shear rate were found to delimit different behaviors of the shear rate-dependencies obtained in this work. (C) 2004 American Institute of Physics.