Many common materials display significant nonlinear theological properties. Characterizing these properties can be done with a variety of methods. One such method uses inertio-elastic oscillations, which occur naturally in rotational rheometry as a consequence of a material's elasticity and the inertia of the rheometer. These oscillations have primarily been used to characterize linear viscoelastic properties. In addition to allowing for the imposition of stress-biased oscillations on short time scales, we demonstrate that extending this technique to nonlinear deformations provides accurate measurements of nonlinear material properties. Our experiments are performed on fibrin networks, which are well characterized and have dramatic nonlinear properties that are biologically significant. We compare the tangent moduli measurements of inertio-elastic oscillations with three standard methods of nonlinear rheology: forced oscillations about a prestress, a geometric interpretation of large amplitude oscillatory shears, and an extension of the linear viscoelastic moduli to the nonlinear regime. Inertio-elastic oscillations provide an accurate characterization of fibrin's nonlinear properties, and further, our measurements suggest that inertio-elastic oscillations provide the most straightforward method of distinguishing between nonlinear elasticity and dissipation at any given stress. In fact, we find that inertio-elastic oscillations provide the most accurate measurement of the subdominant loss component of our networks. (C) 2008 The Society of Rheology.