Granular materials are anomalous transport media. This is due, primarily, to the arrangement of inter-particle contacts: they form contact networks that are (1) heterogeneous, i.e., a few particles support high compressive force, while many others support very little, and (2) self-organized, i.e., spatial correlations are significant, with strong forces forming a sub-network of interconnecting ‘force chains'. Using numerical simulations, we investigate the influence of self-organization on the transport properties of granular matter, with particular attention to heat conduction – a phenomenon of ubiquitous engineering importance. We find that self-organization of the granular network promotes efficient transport. Furthermore, the network response to targeted and random attack suggests that a small minority of interparticle contacts, not necessarily those of highest compressive force, are the primary determinants of transport behavior. A networks-based approach, including concepts such as shortest path and betweenness centrality, provides valuable insight into the observed behavior. It is significant that this approach towards understanding transport in self-organized systems is not limited to the study of granular matter, rather can be extended to an entire class of transport networks exhibiting similar structural motifs, such as vascular networks, automobile transport networks, and flows through porous material.