Primordial nucleosynthesis is calculated in three classes of exact cosmological models possessing anisotropic three-curvature. Although these models do not contain the Friedman isotropic model as a special case and hence cannot isotropize, they lead to primordial helium-4 abundances within the observed range, Y = 0.21-0.26, in many cases. This is unlike the behavior found in the anisotropic cosmological models used in previous nucleosynthesis studies. This difference arises because the curvature anisotropy gives rise to a more complicated expansion anisotropy than in the kasner-like models studied previously. One of the exact models studied possess non-zero vorticity as well as shear and curvature anisotropy. These examples demonstrate that the range of acceptable cosmological models for the early universe may be considerably wider than is usually believed.