The surface impedance characteristics of a wire grid composed of infinitely long parallel conducting wires embedded in lossless and lossy frequency-dispersive chiral background media have been investigated. Using wavefield decomposition approach for a chiral background and with the application of impedance boundary conditions for a wire grid, an analytic expression for the surface impedance of a wire grid with a chiral background has been derived. It is shown that the surface impedance magnitude of a wire grid with chiral nihility background is close to zero and almost independent of incident polar angles. A strong chiral background significantly enhances the surface impedance magnitude of a wire grid for incident polar angles closer to right angle as compared to the free space background. The same electromagnetic appearance of a wire grid with frequency-dispersive chiral and free space background media at some critical frequency has also been discussed which may find applications in electromagnetic illusions. It is also shown that if the value of incident polar angles are closer to right angle then the impedance magnitude of a wire grid embedded in the realistic chiral background is smaller as compared to the same wire grid when placed in the free space background.