The transport of polymers with folded configurations across membrane pores is investigated theoretically by analyzing simple discrete stochastic models. The translocation dynamics is viewed as a sequence of two events: motion of the folded segment through the channel followed by the linear part of the polymer. The transition rates vary for the folded and linear segments because of different interactions between the polymer molecule and the pore. It is shown that the translocation time depends nonmonotonously on the length of the folded segment for short polymers and weak external fields, while it becomes monotonous for long molecules and large fields. Also, there is a critical interaction between the polymers and the pore that separates two dynamic regimes. For stronger interactions, the folded polymer moves slower, while for weaker interactions, the linear chain translocation is the fastest. In addition, our calculations show that the folding does not change the translocation scaling properties of the polymer. These phenomena can be explained by the interplay between translocation distances and transition rates for the folded and linear segments of the polymer. Our theoretical results are applied for analysis of experimental translocations through solid-state nanopores.