The integral equation constrained optimization approach to finding three-dimensional (3D) minimum-drag shapes for bodies translating in viscous incompressible fluid under the Oseen approximation of the Navier-Stokes equations is presented. The approach formulates the Oseen flow problem as a boundary integral equation and finds solutions to this equation and its adjoint in the form of function series. Minimum-drag shapes, being also represented by function series, are then found by the adjoint equation-based method with a gradient-based algorithm, in which the gradient for shape series coefficients is determined analytically. Compared to partial differential equation (PDE) constrained optimization coupled with the finite element method (FEM), the approach reduces dimensionality of the flow problem, solves the issue with region truncation in exterior problems, finds minimum-drag shapes in semianalytical form, and has fast convergence. The approach is demonstrated in three drag minimization problems for different Reynolds numbers for (i) a body of constant volume, (ii) a torpedo with only fore-and-aft noses being optimized, and (iii) a body of constant volume following another body of fixed shape. The minimum-drag shapes in problem (i) are in good agreement with the existing optimality conditions and conform to those obtained by PDE constrained optimization.