We present a general diagrammatic approach to the construction of efficient algorithms for computing the Fourier transform of a function on a finite group. By extending work which connects Bratteli diagrams to the construction of Fast Fourier Transform algorithms we make explicit use of the path algebra connection and work in the setting of quivers. In this setting the complexity of an algorithm for computing a Fourier transform reduces to path counting in the Bratelli diagram, and we generalize Stanley's work on differential posets to provide such counts. Our methods give improved upper bounds for computing the Fourier transform for the general linear groups over finite fields, the classical Weyl groups, and homogeneous spaces of finite groups.