Continuing our research in the field of new heterocyclic compounds, herein we report on the synthesis and antitumor activity of new amino derivatives of pyrido[3′,2′:4,5](furo)thieno[3,2-d]pyrimidines as well as of two new heterocyclic systems: furo[2–e]imidazo[1,2-c]pyrimidine and furo[2,3-e]pyrimido[1,2-c]pyrimidine. Thus, by refluxing the 8-chloro derivatives of pyrido[3′,2′:4,5]thieno(furo)[3,2-d]pyrimidines with various amines, the relevant pyrido[3′,2′:4,5]thieno(furo)[3,2-d]pyrimidin-8-amines were obtained. Further, the cyclization of some amines under the action of phosphorus oxychloride led to the formation of new heterorings: imidazo[1,2-c]pyrimidine and pyrimido[1,2-c]pyrimidine. The possible antitumor activity of the newly synthesized compounds was evaluated in vitro. The biological tests evidenced that some of them showed pronounced antitumor activity. A study of the structure–activity relationships revealed that the compound activity depended mostly on the nature of the amine fragments. A docking analysis was also performed for the most active compounds.