We analyzed the transport of ballistic electrons through organic molecules on uniformly flat surfaces of bismuth grown on silicon. For the fullerene C60 and for a planar organic molecule (3,4,9,10-perylene-tetracarboxylic acid dianhydride), the signals revealed characteristic submolecular patterns that indicated where ballistic transport was enhanced or attenuated. The transport was associated to specific electronic molecular states. At electron energies of a few electron volts, this "scanning near-field electron transmission microscopy" method could be applied to various adsorbates or thin layers.