In the dawn of the Internet of Things (IoT) era, device-to-device interactions on real-time data without involving humans have become a vital part of everyday life. Hence, ultra-low-power sensors and processing units are of the utmost importance. In recent years, micro-electromechanical (MEM) relays have been treated as promising beyond-CMOS candidates due to their zero-leaking and steep turn ON/OFF properties. This paper presents a MEM relay and its characterization, followed by a demonstration of relay-based core logic circuits, including an XOR and an adder. The relays' mechanical operation makes them inevitably slower than transistors; however, this paper demonstrates that a scaled 32-bit relay adder consumes 60 times less energy per operation than its CMOS counterpart in 40 nm technology. The proposed relay circuits, with their ultra-low power consumption property, are particularly suitable for applications with rigorous energy requirements while operating at a slow-to-moderate speed, such as wearable accessories, remote sensors, and implantable biomedical devices.