Lithium solid electrolytes can potentially address two key limitations of the organic electrolytes used in today's lithium-ion batteries, namely, their flammability and limited electrochemical stability. However, achieving a Li+ conductivity in the solid state comparable to existing liquid electrolytes (>1mS cm(-1)) is particularly challenging. In this work, we reveal a fundamental relationship between anion packing and ionic transport in fast Li-conducting materials and expose the desirable structural attributes of good Li-ion conductors. We find that an underlying body-centred cubic-like anion framework, which allows direct Li hops between adjacent tetrahedral sites, is most desirable for achieving high ionic conductivity, and that indeed this anion arrangement is present in several known fast Li-conducting materials and other fast ion conductors. These findings provide important insight towards the understanding of ionic transport in Li-ion conductors and serve as design principles for future discovery and design of improved electrolytes for Li-ion batteries.