The scattering of neutrons with energies of \ensuremath{\lesssim}1.5 MeV from the elements Ta, Re, and Pt was experimentally studied. Elastic and inelastic neutron-scattering angular distributions were determined at incident-neutron-energy intervals of \ensuremath{\lesssim}20 keV throughout the range 0.3 to 1.5 MeV. Time-of-flight measurements were made concurrently at eight or more scattering angles between 25\ifmmode^\circ\else\textdegree\fi{} and 155\ifmmode^\circ\else\textdegree\fi{} with sufficient scattered-neutron-energy resolution to separate the elastically scattered component from most inelastically scattered neutron groups. Differential inelastic scattering cross sections corresponding to the excitation of the following states were observed: in Ta at 144\ifmmode\pm\else\textpm\fi{}10 (doublet), 313\ifmmode\pm\else\textpm\fi{}15, 506\ifmmode\pm\else\textpm\fi{}20, 620\ifmmode\pm\else\textpm\fi{}20, 720\ifmmode\pm\else\textpm\fi{}20, and 930\ifmmode\pm\else\textpm\fi{}25 keV; in Re at 132\ifmmode\pm\else\textpm\fi{}10, 219\ifmmode\pm\else\textpm\fi{}15, 313\ifmmode\pm\else\textpm\fi{}15, 387\ifmmode\pm\else\textpm\fi{}18, 518\ifmmode\pm\else\textpm\fi{}20, 637\ifmmode\pm\else\textpm\fi{}20, 767\ifmmode\pm\else\textpm\fi{}25, 865\ifmmode\pm\else\textpm\fi{}25, 963\ifmmode\pm\else\textpm\fi{}25, 1060\ifmmode\pm\else\textpm\fi{}25, and 1135\ifmmode\pm\else\textpm\fi{}25 keV; and in Pt at 130\ifmmode\pm\else\textpm\fi{}20, 214\ifmmode\pm\else\textpm\fi{}20, 348\ifmmode\pm\else\textpm\fi{}15, 420\ifmmode\pm\else\textpm\fi{}15, 620\ifmmode\pm\else\textpm\fi{}20, 719\ifmmode\pm\else\textpm\fi{}15, 838\ifmmode\pm\else\textpm\fi{}25, and 935\ifmmode\pm\else\textpm\fi{}20 keV. The experimental results were compared with calculations based upon the optical model and the Hauser-Feshbach formula including corrections for resonance-width fluctuations. The experimentally observed level structure is discussed in the context of single-particle and rotational states of deformed nuclei.