We establish residual a posteriori error estimates for lowest-order Raviart-Thomas mixed finite element discretizations of convection-diffusion-reaction equations on simplicial meshes in two or three space dimensions. The upwind-mixed scheme is considered as well and the emphasis is put on the presence of an inhomogeneous and anisotropic diffusion-dispersion tensor and on a possible convection dominance. Global upper bounds in the energy norm for the approximation error are derived, where in particular all constants are evaluated explicitly, so that the estimators are fully computable. Our estimators give local lower bounds for the error as well, hold from the cases where convection or reaction are not present to convection-dominated problems, and their efficiency only depends on local variations in the coefficients and is shown to be optimal as the local Péclet number gets small. The main idea of the proof is a construction of a new scalar variable based on a simple local postprocessing in each element and a subsequent use of the abstract framework arising from the primal weak formulation of the continuous problem. An interesting particular consequence is that the postprocessed variable coincides with the exact solution for one-dimensional pure diffusion problems with piecewise constant coefficients. Numerical experiments confirm the efficiency and robustness of the derived estimators.