The DNA origami technique can enable functionalization of inorganic structures for single-molecule electric current recordings. Experiments have shown that several layers of DNA molecules, a DNA origami plate, placed on top of a solid-state nanopore is permeable to ions. Here, we report a comprehensive characterization of the ionic conductivity of DNA origami plates by means of all-atom molecular dynamics (MD) simulations and nanocapillary electric current recordings. Using the MD method, we characterize the ionic conductivity of several origami constructs, revealing the local distribution of ions, the distribution of the electrostatic potential and contribution of different molecular species to the current. The simulations determine the dependence of the ionic conductivity on the applied voltage, the number of DNA layers, the nucleotide content and the lattice type of the plates. We demonstrate that increasing the concentration of Mg(2+) ions makes the origami plates more compact, reducing their conductivity. The conductance of a DNA origami plate on top of a solid-state nanopore is determined by the two competing effects: bending of the DNA origami plate that reduces the current and separation of the DNA origami layers that increases the current. The latter is produced by the electro-osmotic flow and is reversible at the time scale of a hundred nanoseconds. The conductance of a DNA origami object is found to depend on its orientation, reaching maximum when the electric field aligns with the direction of the DNA helices. Our work demonstrates feasibility of programming the electrical properties of a self-assembled nanoscale object using DNA.