We combine a number of powerful mathematical techniques to produce an effective method for treating multidimensional radiative transfer problems in complex atomic models without assuming local thermodynamic equilibrium (LTE). The approach is so efficient that multilevel two-dimensional (2D) modeling can now be performed with no more than a workstation. We employ Accelerated Lambda Iteration (ALI) methods: accurate short characteristics for the formal solution of the transfer equation with an efficient new strategy for horizontal periodic boundary conditions, local approximate Lambda-operators given by the diagonal of the exact operator, methods to accelerate the convergence, and preconditioning of the rate equations. Of particular interest is a simple grid-doubling strategy which both rapidly finds the converged solution in very fine meshes and also estimates the true error of that solution. The properties of the method are described in detail with the help of 2D line-transfer calculations with multilevel model atoms for Ca II and H. These illustrative multilevel calculations in schematic inhomogeneous atmospheres demonstrate the importance of properly including the effects of horizontal radiative transfer and realistic atomic models.