This research resolves a long-standing problem on the electromagnetic response of self-complementary metallic screens with checkerboardlike geometry. Although Babinet's principle implies that they show a frequency-independent response, this unusual characteristic has not been observed yet due to the singularities of the metallic point contacts in the checkerboard geometry. We overcome this difficulty by replacing the point contacts with resistive sheets. The proposed structure is prepared and characterized by terahertz time-domain spectroscopy. It is experimentally confirmed that the resistive checkerboard structures exhibit a flat transmission spectrum over 0.1-1.1 THz. It is also demonstrated that self-complementarity can eliminate even the frequency-dependent transmission characteristics of resonant metamaterials.