The tools commonly used to understand structure-property relationships in molecular conductance, inter-atomic currents and conductance eigenchannels, generally give us a sense of familiarity, with the chemical bonding framework and molecular orbitals reflected in the current. Here we show that while this picture is true for conjugated molecules, it breaks down in saturated systems. We investigate the current density in saturated chains of alkanes, silanes and germanes and show that the current density does not follow the bonds, but rather the nuclei define the diameter of a pipe through which the current flows. We discuss how this picture of current density can be used to understand details about the electron transport properties of these molecules. Understanding the spatial distribution of current through molecules, rather than simply the magnitude, provides a powerful tool for chemical insight into physical properties of molecules that are related to current flow.