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Understating elastic properties of thin films is a matter of deep interest both in scientific and industrial fields. In this paper, we propose a combination of resonant-ultrasound spectroscopy coupled with the laser-Doppler interferometry and pump-probe photoacoustics for measuring anisotropic elastic constants of deposited thin films. Among the five independent elastic constants, the resonant-ultrasound spectroscopy is sensitive to the in-plane elastic constants, C11, C13, and C66, and pump-probe photoacoustics to the out-of-plane elastic constant, C33. We apply this to Cu thin films deposited on monocrystalline Si substrates by the magnetron-sputtering method. Cu thin films show elastic anisotropy, C33>C11. The cause of this elastic anisotropy is attributed to the textured structure and the columnar structure. These effects are estimated by X-ray diffraction measurements and micromechanics calculations.