A design method is proposed for a nonlinear disturbance observer based on the notion of passivity. As an initial application, we consider here systems whose structure comprises a set of integrator cascades, though the proposed approach can be extended to a larger class of systems. We describe an explicit procedure to choose the output of the system and to design the nonlinear feedback law used by the observer, provided the system satisfies a sufficient condition for output feedback semi-passification. The output injection term in the observer scales the measurement residual with a nonlinear gain that depends on the output and a set of static design parameters. We provide guidance for parameter tuning such that the disturbance tracking performance and the transient response of the estimation error can be intuitively adjusted. Example applications to two nonlinear mechanical systems illustrate that the proposed nonlinear observer design method is quite effective, producing an observer that can estimate a wide range of disturbances without any need to know or assume the disturbance dynamics.