Efficient and robust iterative solvers for strongly anisotropic elliptic equations are very challenging. Indeed, the discretization of this class of problems gives rise to a linear system with a condition number increasing with anisotropic strength. This weakness is addressed clearly by adopting the asymptotic-preserving (AP) discretizations. In this report a block preconditioning method is introduced to solve the linear algebraic systems of a class of micro–macro asymptotic-preserving (MMAP) scheme. The MMAP method was developed by Degond et al. in 2012 where its corresponding discrete matrix has a 2 × 2 block structure. Motivated by approximate Schur complements, a series of block preconditioners are constructed. With these block preconditioners, a preconditioned GMRES iterative method is developed to solve the discrete equations. Several numerical tests show that block preconditioning methods can be a practically useful strategy with respect to grid refinement and anisotropic strengths. We then give some ideas to develop robust multigrid algorithm for this type of problem.