We carried out tight-binding electronic band structure calculations for the three representative types of ternary platinum oxides with columnar PtO4 chains, NaPt3O4, CaPt2O4, and CdPt3O6, and for LaPd3S4, the sulfur analog of NaPt3O4. As far as the z(2) bands are concerned, the three orthogonal columnar PtO4 chains of NaPt3O4 are nearly independent, but NaPt3O4 does not undergo a metal-to-insulator transition. This implies that the lattice of NaPt3O4 is too stiff to accommodate a periodic lattice distortion associated with the partially filled z(2) bands. As far as the x(2) - y(2) bands are concerned, however, the three orthogonal columnar TX(4) chains of the MT(3)X(4) lattice (T = Pt, X = O; T = Pd, X = S) are not independent. Due to the through-bond sigma interactions that occur between the columnar TX(4) chains, the partially-filled x(2) - y(2) bands are wide and possess 3D character. This explains why (RE)Pd3S4 is metallic. Despite the dimerization in its columnar PtO4 chains, CaPt2O4 is metallic because the upper half of the z(2) band overlaps with other d-block bands. The dimerization of the columnar PtO4 chains does not represent a charge density wave instability but is probably caused by the alternation of filled and unfilled square-prism sites along each chain. The very small band gap of CdPt3O6 (0.04 eV) does not represent the energy separation between the z(2) and x(2) - y(2) bands but may originate from holes introduced into the z(2) band by nonstoichiometry and counterion disorder.