Green's functions lie at the foundation of many uncertainty quantification and uncertainty reduction techniques (e.g., the moment differential equation approach, parameter and/or source identification, and data assimilation). We discuss an accurate and numerically efficient approach to compute Green's functions for transport processes in heterogeneous composite media. We focus on elliptic partial differential equations with (random) discontinuous coefficients. The approach relies on a regularization technique to obtain an associated regular problem, which can be solved using standard finite element methods. We perform numerical experiments to assess the performance of the regularization approach and to evaluate the effects of strong coefficient discontinuities on the Green's function behavior.