Understanding the mechanisms of lithium-ion transport in polymers is crucial for the design of polymer electrolytes. We combine modular synthesis, electrochemical characterization, and molecular simulation to investigate lithium-ion transport in a new family of polyester-based polymers and in poly(ethylene oxide) (PEO). Theoretical predictions of glass-transition temperatures and ionic conductivities in the polymers agree well with experimental measurements. Interestingly, both the experiments and simulations indicate that the ionic conductivity of PEO, relative to the polyesters, is far higher than would be expected from its relative glass-transition temperature. The simulations reveal that diffusion of the lithium cations in the polyesters proceeds via a different mechanism than in PEO, and analysis of the distribution of available cation solvation sites in the various polymers provides a novel and intuitive way to explain the experimentally observed ionic conductivities. This work provides a platform for the evaluation and prediction of ionic conductivities in polymer electrolyte materials.