In this paper, we study the effects of the interparticle interaction range on heat flow. We show that, by increasing the interaction range, we may amplify the thermal conductivity and even change the regime of heat transport. More importantly, considering a crucial problem of phononics, namely, the search of a suitable thermal diode, we investigate the range effects in some graded systems in which thermal rectification is a ubiquitous phenomenon. In such graded models, we show that long-range interactions may significatively increase the rectification power and may avoid its decay with the system size, thus solving relevant problems of the usual proposals of rectifiers. Our results indicate that graded materials are genuine candidates for the actual fabrication of thermal diodes.