An approximate analytical method is described for determining the sound produced by a class of complex fluid-structure interactions in low Mach number flows. This can be used to model noise sources in practical systems, and to check the accuracy of predictions based on time-accurate numerical solutions of the Navier-Stokes equations. The dominant acoustic sources are dipoles whose strengths are dependent on the unsteady surface forces, and are expressed in terms of fluid velocity and vorticity, and a set of harmonic functions determined by the shapes of the structural elements interacting with the flow. (The theory of surface forces for arbitrary motion of a rigid body in viscous, incompressible flow in the presence of a fixed system of boundaries is discussed in an appendix.) These elements can influence both the generation and propagation of sound, and are frequently sources of new vorticity shed into wakes. The procedure is illustrated by application to a model problem in which sound is generated by a vortex interacting with a shrouded rotor in a duct. High-frequency sound is generated when the vortex is drawn into the rotor disc and `chopped' by the blades. Sound is also produced through indirect blade-vortex interactions which, in this case, occur as a result of unsteady blade loadings produced when the core of the vortex is close to the leading edge of the shroud. This is relatively low-frequency sound and is the only component of blade-vortex interaction noise when the vortex is convected through the gap between the shroud and wall of the duct.