The problem of minimizing multiple integral functionals with nonquasiconvex integrands is considered. A numerical method, which is based on an alternative minimizing problem to the relaxed problem and thus uses no quasiconvex envelope of the integrands nor its numerical approximation in the computation, is introduced to approximate simultaneously the highly oscillating minimizing sequences, or in other words microstructures, and the minimizers of the corresponding relaxed problem. Existence and convergence of the discrete solutions are proved and an error estimate is obtained. A numerical example is given.