Effcient iterative methods for the numerical solution of three-dimensional acoustic scattering problems are considered. The original exterior boundary value problem is approximated by truncating the unbounded domain and by imposing a non-reflecting boundary condition on the artificial boundary. The finite element discretization of the approximate boundary value problem is performed using locally fitted meshes, and the mesh equations are solved with algebraic fictitious domain methods with separable preconditioners. These methods are based on imbedding the original domain into a larger domain with a simple geometrical form, which in this work is a sphere or a parallelepiped. The iterative solution method is realized in a low-dimensional subspace, and partial solution technique is applied to the linear systems with the preconditioner. Results of several numerical experiments demonstrate the efficiency and accuracy of the approach.