When a Fourier transform (FT) is to be numerically computed and subsequently numerically inverted, perhaps repetitively, it is desirable that the algorithm used to accomplish this should maintain the orthogonal nature of the Fourier expansion. Algorithms with this property are derived here for the FT of central functions in one, two, and three dimensions. These rules for mechanical quadrature are similar to the trapezoidal rule, but with intervals determined by the zeros of the orthogonal basis functions. A numerical test using Gaussians shows a high accuracy in the computed transforms. The procedure can be extended to other Fourier-Bessel transforms.