The wide range of chemical compositions exhibited by polymers enables the fabrication of materials having highly tunable cohesive energy strength epsilon, and many of the properties that make polymers so useful as structural and responsive materials in both manufacturing and living systems derive from the variability of this basic property. The design and characterization of polymer materials then inevitably leads to a consideration of how e impacts the thermodynamic and relaxation properties of polymer liquids. Our prior paper uses molecular dynamics simulations of a model coarse-grained polymer melt to systematically investigate the dependence of commonly measured thermodynamic properties on e, while the present work focuses on the relaxation dynamics of the same molecular model. After demonstrating, as expected, that e greatly influences the segmental relaxation time, we obtain a universal reduction of all our data for relaxation in terms of an activated transport model in which the activation free energy is increased from its high temperature value by a factor precisely determined by the average extent of the cooperative motion of monomers in the polymer liquid. This data reduction is consistent with the recently developed string model of glass formation, as well as with the assumptions of the generalized entropy theory of glass formation derived from a combination of the classical Adam-Gibbs model with a statistical mechanical model of polymer melts. In addition to providing firm observational data facilitating the development of improved theories of polymer glass formation, our results also yield insights into the molecular origin of particular thermodynamic and relaxation properties of polymers, insights that should aid in the design of polymer materials.