A generalized formulation of a boundary integral equations method is presented for analysis of forced vibrations of a composite elastic structure immersed in compressible inviscid fluid. The structure is supposed to consist of parts which are membranes, plates, spherical, conical or cylindrical shells. Both the interaction between the acoustic medium and the composite structure as a whole, and the interactions between the parts of the structure, are described by boundary integral equations. These boundary integral equations are assembled in a two-level system. The first level boundary integral equations govern the dynamics of the above-mentioned ''simple'' parts of the structure. They contain unknown boundary displacements and forces, contact acoustic pressure and driving loads. The kernels of these equations are Green functions of ''simple'' unbounded structures vibrating in vacuo. These functions have explicit analytical forms. The boundary integral equation of the second level governs the interaction between the fluid and the structure as a whole. The classical boundary integral equation related to the contact acoustic pressure is modified by substitution of Somigliana-type formulae for normal displacments on each part of the structure. As a result the second level equation constitutes the connection between the contact acoustic pressure, driving loads and the boundary displacements and forces on the edges of the simple parts of the structure. The kernels of this boundary integral equation are convolutions of Green functions of simple unbounded structures and unbounded fluid. The validity of the method proposed is demonstrated for several simple test problems analyzed earlier by other authors. The aim of this part of the paper is to outline and evaluate the numerical procedure. A detailed analysis of the equations of vibrations of a composite thin-walled structure in an acoustic medium is presented in Part 2 of this paper.