We prove the Plancherel formula for a four-parameter family of discrete Fourier transforms and their multivariate generalizations stemming from corresponding generalized Schur polynomials. For special choices of the parameters, this recovers the sixteen classic discrete sine- and cosine transforms DST-1,...,DST-8 and DCT-1,...,DCT-8, as well as recently studied (anti-)symmetric multivariate generalizations thereof.