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Dispersion diagrams and magnetic- and electric-field distributions are found for a microstrip structure containing a simple left-handed medium (LHM). The LHM material constituting the substrate has characteristics chosen to overlap with potentially realizable substances. Calculations are done using a fast solver anisotropic Green's function spectral-domain computer code. The theoretical-field method, valid for complex layered structures and anisotropies in the LHM, underlying the computational procedure, which is critical for finding the dispersion diagrams and field distributions, is presented. It is found that the dispersion diagrams and field distributions are very unusual and admit the possibility of completely new device realizations based on combining LHM with conventional materials in a multilayered configuration.