Rheology and microstructure of plate-like particle suspensions in linear shear flows are studied through numerical simulations for a range of volume fraction phi up to 0.30. Particles with aspect ratio AR= 3-7 are modeled as planar assemblages of spheres. Numerical methods are developed to calculate the hydrodynamic interactions based on an extension of the Stokesian dynamics method for spheres. At low phi, suspensions exhibit a degree of particle alignment consistent with the orientation distribution predicted by Jeffery orbits. At high phi, hydrodynamic interactions produce a high degree of ordering with particles aligned in horizontal layers perpendicular to the gradient direction. This allows sufficient free volume for shearing of the suspension with moderate viscosity at high phi. A second ordering mechanism is the formation of transient stacks of plate-like particles which move as rigid assemblies. The two mechanisms - particle alignment and particle stacking reduce the effects of hydrodynamic interaction. Over the range of phi considered, viscosity is not a strong function of aspect ratio for non-Brownian suspensions, because the increased hydrodynamic interactions of high aspect ratio particles are offset by the increased degree of particle ordering. (C) 2008 The Society of Rheology.