Thin films of aligned nanowires composed of oriented para-sexiphenyl molecules were prepared on muscovite mica substrates and transferred to the surfaces of Ag films. Results of attenuated total reflection (ATR) measurements on these samples with a fixed wavelength of incident light clearly demonstrate the anisotropic propagation of surface plasmon polaritons (SPPs) induced by the optical anisotropy in the nanowire films. The magnitude of the SPP wave vector takes a maximum value when the SPP propagates along the direction of molecular orientation. Results of ATR measurements performed for various wavelengths of the incident light reveal the anisotropy in the SPP dispersion curve in a wide range of photon energy. From the fitting of theoretical reflection curves to observed ATR spectra, dispersions of the principal values of dielectric tensor of the nanowire film are obtained.