A Boundary Integral Equation (BIE) method is presented which efficiently computes the harmonic acoustic response of axi-symmetric structures. The key idea is that the BIE's are Fourier transformed via FFT's in the azimuthal variable; this reduces the azimuthal periodic convolutions to simple multiplications. The original three-dimensional problem is transformed into a series of M decoupled two-dimensional problems, where M is the number of azimuthal Fourier components. If N is the number of nodal points along the semi-perimeter of a scatterer, then the harmonic response can be computed with just O(N3M) algebraic operations. This is far less expensive than solving the original problem which requires O(N6) algebraic operations per frequency. Moreover, the active memory requirement is reduced from O(N4) to O(N2) complex words, and the algorithm is ideally suited to a parallel computer. Examples are given where less than three minutes were required by a Gould computer (4 MIPs) to compute the harmonic response of a scatterer four wavelengths in dimension.