The convergence properties of multigrid algorithms are defined by two factors: (1) the smoothing rate which describes the reduction of high-frequency error components and (2) the quality of the coarse-grid correction which is responsible for dumping of smooth error components. In elliptic problems, all the fine-grid smooth components are well approximated on the coarse grid built by standard (full) coarsening.  In nonelliptic problems, however, some fine-grid  components that are much smoother in the characteristic direction than in other directions, cannot be approximated with standard multigrid methods. We present a novel multigrid approach to the solution of nonelliptic problems. This approach is based on semicoarsening and well-balanced explicit correction terms, added to coarse-grid operators to maintain on coarse grids the same cross-characteristic interaction as on the target (fine) grid. Multicolor relaxation schemes are used on all the levels, allowing a very efficient parallel implementation. Applications to the 2-D constant-coefficient convection operator discretized on vertex and cell centered grids are demonstrated. The resulting multigrid algorithms demonstrate the ``textbook multigrid efficiency'', meaning the solution to the governing problem is attained in a computational work which is a small multiple of the operation count in the discretized problem itself. Extensions to the three dimensions, to variable coefficients, and to convection-diffusion problems are discussed.