The formation of one-dimensional band structures has been observed for Kr and Xe adsorbed on the reconstructed Pt(110)1 x 2 surface at submonolayer coverages: The p3/2 and p1/2 valence levels of both adsorbates exhibit significant dispersion for the [11-bar0] direction, i.e., along the densely packed substrate rows, but no dispersion is observed for [001], i.e., perpendicular to the rows. The observed k-space periodicities and tight-binding band-structure calculations imply that both Kr and Xe are adsorbed in densely packed linear chains on the reconstructed Pt(110)1 x 2 surface, with negligible interaction between neighboring rows.