A minimization problem modeling geometrically exact generalized continua of micromorphic type is considered. The solution consists of two fields, the elastic deformation φ of a given body and a tensorial field P which can model different additional features needed for a more reliable description of solids. For the solution of this minimization problem, a staggered algorithm is introduced which decouples the original problem into two separate problems. In each of these subproblems, one of the variables, φ or P, respectively, is kept fixed and the subproblem is solved for the remaining variables, i.e., P or φ, respectively. Each of the problems is discretized with finite elements. Numerical results are presented for a cubic and a cylindrical geometry.