This paper deals with the broadband noise due to the interaction between convected turbulent gusts and a rectilinear cascade of flat plates bounded by two parallel walls. An analytic formulation for the acoustic power spectrum due to this turbulence-cascade interaction is derived, which can be used to assess the effects of the span-wise wavenumber components of ingesting turbulent gust on the overall acoustic power spectrum. This three-dimensional theory is based on the two-dimensional theory of Cheong et al. (2006, 2009). The three-dimensional model is shown to provide a close fit to the measured spectrum of rotor-stator interaction. The predictions using this three-dimensional model are also compared with those using the previous two-dimensional model by Cheong et al. (2006; 2009). Through this comparison, it is found that the contributions to the acoustic power of the span-wise wavenumber components of incident turbulent gust are increased as the frequency is augmented, which is mainly due to three-dimensional dispersion-relation characteristics of acoustic waves. This implies that the number of incident turbulent gust modes directly involved in generating cut-on acoustic waves increases as the frequency increases. Therefore, in the lower frequency range, three-dimensional acoustic power is less than its corresponding two-dimensional one, whereas, as the frequency increases, the three-dimensional acoustic power spectrum closely follows those of two-dimension. The formulation is also used to make a parametric study about the effects on the power spectrum of the blade number, stagger angle, gap-chord ratio, and Mach number.