Background: The kidney exchange problem (KEP) addresses the matching of patients in need for a replacement organ with compatible living donors. Ideally many medical institutions should participate in a matching program to increase the chance for successful matches. However, to fulfill legal requirements current systems use complicated policy-based data protection mechanisms that effectively exclude smaller medical facilities to participate. Employing secure multi-party computation (MPC) techniques provides a technical way to satisfy data protection requirements for highly sensitive personal health information while simultaneously reducing the regulatory burdens. Results: We have designed, implemented, and benchmarked SPIKE, a secure MPC-based privacy-preserving KEP which computes a solution by finding matching donor-recipient pairs in a graph structure. SPIKE matches 40 pairs in cycles of length 2 in less than 4 minutes and outperforms the previous state-of-the-art protocol by a factor of 400x in runtime while providing medically more robust solutions. Conclusions: We show how to solve the KEP in a robust and privacy-preserving manner achieving practical performance. The usage of MPC techniques fulfills many data protection requirements on a technical level, allowing smaller health care providers to directly participate in a kidney exchange with reduced legal processes.