Thermoplastic polyurethanes constitute a versatile family of materials with a broad variety of engineering applications. However, connection between their chemical structure and mechanical properties remains elusive, in large part due to their heterogeneous nature, arising from segregation of chemically distinct segments into separate domains, with resulting complex morphologies. Using atomistic simulations, we examine the structure and mechanical properties of a common family of thermoplastic polyurethanes (TPU) comprising 4,4'-diphenylmethane diisocyanate and n-butanediol (hard segment) and poly(tetramethylene oxide) (soft segment). A lamellar stack model previously developed for the study of semicrystalline polymers is applied here for the first time to a phase-segregated copolymer. Equilibrium structure and properties were evaluated for TPUs with different ratios of hard and soft components, using a combination of Monte Carlo and molecular dynamics simulations. Stress-strain behaviors were then evaluated using nonequilibrium molecular dynamics (NEMD) simulations. The compositional dependence of the Young's moduli thus obtained is shown to be well-approximated by a micromechanical homogenization model of the hard and soft components. Voigt (upper) and Reuss (lower) bounds of modulus were obtained for orientationally averaged aggregates and shown to be greater than those measured experimentally. The discrepancy is explained in terms of the strain rate dependence of elastic moduli, characterized by an Eyring-like function.