Applications of face recognition systems for authentication purposes are growing rapidly. Although state-of-the-art (SOTA) face recognition systems have high recognition accuracy, the features which are extracted for each user and are stored in the system's database contain privacy-sensitive information. Accordingly, compromising this data would jeopardize users' privacy. In this paper, we propose a new cancelable template protection method, dubbed MLP-hash, which generates protected templates by passing the extracted features through a user-specific randomly-weighted multi-layer perceptron (MLP) and binarizing the MLP output. We evaluated the unlinkability, irreversibility, and recognition accuracy of our proposed biometric template protection method to fulfill the ISO/IEC 30136 standard requirements. Our experiments with SOTA face recognition systems on the MOBIO and LFW datasets show that our method has competitive performance with the BioHashing and IoM Hashing (IoM-GRP and IoM-URP) template protection algorithms. We provide an open-source implementation of all the experiments presented in this paper so that other researchers can verify our findings and build upon our work.