We propose a theory of the origin of transport nonuniversality in disordered insulating-conducting compounds based on the interplay between microstructure and tunneling processes between metallic grains dispersed in the insulating host. We show that if the metallic phase is arranged in quasi- one-dimensional chains of conducting grains, then the distribution function of the chain conductivities g has a power-law divergence for g-->0, leading to nonuniversal values of the transport critical exponent t. We evaluate the critical exponent t by Monte Carlo calculations on a cubic lattice, and show that our model can describe universal as well nonuniversal behaviors of transport depending on the value of few microstructural parameters. Such a segregated tunneling-percolation model can describe the microstructure of a quite vast class of materials known as thick-film resistors, which display universal or nonuniversal values of t depending on the composition.