Electronic structures of three superconducting rare-earth iron silicides (Lu;Y;Sc)2Fe3Si5 and nonsuperconducting Lu2Ru3Si5, adopting a tetragonal crystal structure (P4/mnc), have been calculated employing the full-potential local-orbital method within the density functional theory. The investigations were focused particularly on the band structures and Fermi surfaces, existing in four bands and containing rather threedimensional electronlike and holelike sheets. They support an idea of unconventional multi-band superconductivity in these ternaries, proposed earlier by other authors for Lu2Fe3Si5, based on heat-capacity, resistivity, electromagnetic and muon spin rotation measurements. Finally, a discussion on differences in the electronic structures between the investigated here and other common families of iron-based superconductors is carried out.