Theoretical results on the electromagnetic wave diffraction from a periodic strip grating placed on a chiral medium are obtained. Analytical Regularization Method based on the solution to the vector Riemann-Hilbert boundary value problem was used to get robust numerical results in the resonant domain, where direct solution methods typically fail. It was shown that in the case of normal incidence of linearly polarized wave the cross-polarized Field appears in the reflected Field. For elliptically polarized incident wave the diffraction character essentially depends on the polarization direction of the incident wave. These diffraction peculiarities are more pronounced in the resonant domain. Influence of the dichroism caused by chiral medium losses is thoroughly studied. The combination of a chiral medium and a grating can be effectively used for a frequency and polarization selection and for a mode conversion.