We present the enhancement of frequency doubled and tripled light in zinc oxide (ZnO) by utilizing arrays of Mie-resonant dielectric nanoantennas. The cylindrical antennas exhibit resonances at the fundamental excitation wavelength, enabling strong localization of light and redistribution of the electric field inside the structures. Linear transmission spectroscopy reveals the resonant behavior of the fabricated nanoantennas and confirms the performed theoretical simulations. Nonlinear spectroscopic measurements show direct dependency between the resonant behavior and the amplification of nonlinear signals. Comparing the unpatterned ZnO thin films with the dielectric nanostructures shows a maximum enhancement by factors of about 22 for SHG and 7 for THG from resonant excitations of the nanoantennas.