Model liquids have been constructed to study the role of local structure in the anomalous properties of liquid water. The intermolecular potentials were modified by increasing the weight of the Lennard-Jones term relative to the electrostatic term in the SPC/E model for water. The resulting family of liquids varies from SPC/E water to a Lennard-Jones-like liquid. Properties were measured as a function of density and temperature. The local structure was described by two order parameters, one measuring the tetrahedral order and the other measuring the translational order. The translational order parameter was found to be large for both tetrahedral and Lennard-Jones liquids, but to go through a minimum as the potentials were modified, demonstrating that the two types of structure are incompatible. Just as in water several properties (e.g., the translational diffusion coefficient, entropy) exhibit anomalous density dependence as a result of the breakdown of local tetrahedrality, we observed nonmonotonic behavior of the translational diffusion constant and reorientational relaxation rate as the fluids were transformed from tetrahedral to Lennard-Jones-like. This is also an indication of the incompatibility between Lennard-Jones and water-like structure.