  We display some simple cosmological solutions of gravity theories with
quadratic Ricci curvature terms added to the Einstein-Hilbert lagrangian which
exhibit anisotropic inflation. The Hubble expansion rates are constant and
unequal in three orthogonal directions. We describe the evolution of the
simplest of these homogeneous and anisotropic cosmological models from its
natural initial state and evaluate the deviations they will create from
statistical isotropy in the fluctuations produced during a period of
anisotropic inflation. The anisotropic inflation is not a late-time attractor
in these models but the rate of approach to a final isotropic de Sitter state
is slow and is conducive to the creation of observable anisotropic statistical
effects in the microwave background. The statistical anisotropy would not be
scale invariant and the level of statistical anisotropy will grow with scale.
