We present a novel all-dielectric left-handed metamaterial (LHM) in the X-band microwave regime. The LHM is formed by cubes and square rods, which are made of microwave ceramic with high permittivity and low loss. Both Mie theory and dielectric resonator theory are found to play important roles in generating negative effective parameters. The effective permeability is negative in the designed frequency range due to the first Mie resonance associated with the cubes, while the effective permittivity is negative below the plasma frequency owing to the second resonance mode of the square rods. The parameter retrieval method, transmission spectra method and wedge-shaped negative refraction method are used to demonstrate that the composite has a double-negative regime. Experiments are carried out to verify the designed all-dielectric LHM. The proposed method presents new ways to realize all-dielectric LHMs.