In this work we develop a unifying and general method for calculating linear viscoelastic properties of multicomponent colloidal mixtures of spherical particles. Using linear response theory based on the many-body Smoluchowski diffusion equation, we derive an exact expression for the zero shear rate shear relaxation function, together with a Green-Kubo formula for the static shear viscosity. From these results, we obtain an exact expression for the high frequency elastic shear modulus of colloidal mixtures. We present, in addition, the first derivation of a self-consistent mode coupling scheme for the linear viscoelasticity of concentrated colloidal mixtures. This scheme offers the opportunity for a unified description of linear viscoelasticity and diffusion mechanisms, It accounts further for polydispersity and mixing effects, and leads naturally to a diverging shear viscosity at a glass transition point. Various limiting cases are considered to assess the accuracy of the approach. It is-shown to be a valuable method for evaluating the rheological properties of concentrated colloidal mixtures.