In this study, we evaluated the magnetization properties of a magnetic alloy with single-crystalline cubic nanostructures, in order to clarify its magnetocrystalline anisotropy. Upon applying a specific annealing treatment to the CuNiFe base material, the precipitated magnetic particles grew into cubic granules, resulting in the formation of nanometric cubic single crystals of magnetic CuNiFe in a nonmagnetic Cu-rich matrix. The cubic nanostructures of CuNiFe were oriented along their crystallographic axis, in the direction of the face-centered-cubic structure. We evaluated the static magnetization properties of the sample, which originated primarily from the CuNiFe nanocubes precipitated in the Cu-rich matrix, under an applied DC magnetic field. The magnetocrystalline anisotropy was readily observed in the magnetization curves. The axis of the CuNiFe was observed to be the easy axis of magnetization. We also investigated the dynamic magnetization properties of the sample under an AC magnetic field. By subtracting the magnetic signal induced by the eddy current from the magnetization curves of the sample, we could obtain the intrinsic AC magnetization properties of the CuNiFe nanocubes.