This review provides results obtained by scientists from different countries on the antiviral activity of medicinal mushrooms against influenza viruses that can cause pandemics. Currently, the search for antiviral compounds is relevant in connection with the coronavirus disease 2019 (COVID-19) pandemic caused by severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2). Medicinal mushrooms contain biologically active compounds (polysaccharides, proteins, terpenes, melanins, etc.) that exhibit an antiviral effect. The authors present the work carried out at the State Research Center of Virology and Biotechnology Vector in Russia, whose mission is to protect the population from biological threats. The research center possesses a collection of numerous pathogenic viruses, which allowed screening of water extracts, polysaccharides, and melanins from fruit bodies and fungal cultures. The results of investigations on different subtypes of influenza virus are presented, and special attention is paid to Inonotus obliquus (chaga mushroom). Compounds produced from this mushroom are characterized by the widest range of antiviral activity. Comparative data are presented on the antiviral activity of melanin from natural I. obliquus and submerged biomass of an effective strain isolated in culture against the pandemic strain of influenza virus A/California/07/09 (H1N1 pdm09).