Lithium-ion secondary batteries are commonly used to power many consumer devices such as handheld phones, laptops, portable music players, and even electric vehicles. One of the key properties that influence the performance of lithium-ion batteries is the ionic conductivity of the electrolyte (i.e., the movement of Li ions from one electrode to another). This is dependent on the speed at which Li ions diffuse across the cell and related to the solvation structure of the Li ions. The choice of solvent can greatly impact both solvation and diffusivity of Li ions. In this work, we use first principles molecular dynamics to examine the solvation and diffusion of lithium ions in the bulk organic solvents ethylene carbonate (EC), ethyl methyl carbonate (EMC), and a mixture of EC/EMC. We find that Li ions are solvated by either carbonyl or ether oxygen atoms of the solvents and sometimes by the PF₆ anion. Li⁺ prefers a tetrahedrally-coordinated first solvation shell regardless of which species are involved, with the specific preferred solvation structure dependent on the organic solvent. In addition, we calculate Li diffusion coefficients in each electrolyte, finding slightly larger diffusivities in the linear carbonate EMC compared to the cyclic carbonate EC. The magnitude of the diffusion coefficient correlates with the degree of Li⁺ solvation.  Corresponding analysis for the PF₆ anion shows greater diffusivity associated with a weakly-bound, poorly defined first solvation shell. Results from this work can be used to design new bulk electrolytes that will improve the performance of current Li-ion batteries.