This paper presents a system identification technique for systems whose output is asymptotically periodic under constant inputs. The model used for system identification is a discrete-time Lur'e model consisting of asymptotically stable linear dynamics, a time delay, a washout filter, and a static nonlinear feedback mapping. For all sufficiently large scalings of the loop transfer function, these components cause divergence under small signal levels and decay under large signal amplitudes, thus producing an asymptotically oscillatory output. A bias-generation mechanism is used to provide a bias in the oscillation. The contribution of the paper is a least-squares technique that estimates the coefficients of the linear model as well as the parameterization of the continuous, piecewise-linear feedback mapping.