A new lattice-gas cellular automaton model for simulating binary fluids in three dimensions is introduced. It is particularly suitable for modeling slow flows of mixtures with complicated interface geometries or within complicated boundaries, such as in the interior of a porous rock. Phase separation is triggered spontaneously in the model by statistical fluctuations and phase domains are approximately isotropic. The measured surface tension is large compared to that in analogous two-dimensional models. The model is applied to a study of the time-dependent effective viscosity of a phase-separating mixture in a simple shear flow. Results qualitatively match both experiment and theory: the viscosity increases rapidly, then decays gradually to a steady-state value which is larger than the viscosity of the pure fluids. The effective viscosity increases with increasing concentration and decreases with increasing strain rate.