We report on the synthesis of ZnO nanoplates and ZnO nanoplate/Ag nanoparticle heterostructures via a simple and cost effective wet chemical precipitation method. The prepared samples were characterized for structural and optical properties by X-ray diffraction (XRD), Scanning Electron Microscopy (SEM), UV-VIS reflectance, Raman, and FT-IR spectroscopy. The Kramers-Kronig (K-K) method and classical dispersion theory was applied to calculate the far-infrared optical constants such as, refractive index n(ω), dielectric constant ε(ω), transverse optical phonon (TO) and longitudinal (LO) optical phonon modes. We determined various optical constant values n(ω) and ε(ω) for ZnO nanostructures in the range of 0 to 9 and 0 to 70, respectively. Whereas, for Ag deposition on ZnO nanostructures, the corresponding n(ω) and ε(ω) values were found to be increased in the range of 0 to 30 and 0 to 800, respectively. The TO and LO optical phonon modes of ZnO nanoplate/Ag nanoparticle heterostructures were also found to be higher (416 cm−1, 620 cm−1) when compared with corresponding values obtained for ZnO nanoplates (415 cm−1, 604 cm−1).