The motion of polymers with inhomogeneous structure through nanopores is discussed theoretically. Specifically, we consider the translocation dynamics of polymers consisting of double-stranded and single-stranded blocks. Since only the single-stranded chain can go through the nanopore the double-stranded segment has to unzip before the translocation. Utilizing a simple analytical model, translocation times are calculated explicitly for different polymer orientations, i.e., when the single-stranded block enters the pore first and when the double-stranded segment is a leading one. The dependence of the translocation dynamics on external fields, energy of interaction in the double-stranded segment, size of the polymer, and the fraction of double-stranded monomers is analyzed. It is found that the order of entrance into the pore has a significant effect on the translocation dynamics. The theoretical results are discussed using free-energy landscape arguments.