We introduce an implicit solvent Molecular Dynamics approach for calculating ionic fluxes through narrow nanopores and transmembrane channels. The method relies on a dual-control-volume grand-canonical molecular dynamics (DCV-GCMD) simulation and the analytical solution for the electrostatic potential inside a cylindrical nanopore recently obtained by Levin [Europhys. Lett. 76, 163 (2006)]. The theory is used to calculate the ionic fluxes through an artificial transmembrane channel which mimics the antibacterial gramicidin A channel. Both current-voltage and current-concentration relations are calculated under various experimental conditions. We show that our results are comparable to the characteristics associated to the gramicidin A pore, especially the existence of two binding sites inside the pore and the observed saturation in the current-concentration profiles.