There have been significant advances in the field of Internet of Things (IoT) recently. At the same time there exists an ever-growing demand for ubiquitous healthcare systems to improve human health and well-being. In most of IoT-based patient monitoring systems, especially at smart homes or hospitals, there exists a bridging point (i.e., gateway) between a sensor network and the Internet which often just performs basic functions such as translating between the protocols used in the Internet and sensor networks. These gateways have beneficial knowledge and constructive control over both the sensor network and the data to be transmitted through the Internet. In this paper, we exploit the strategic position of such gateways to offer several higher-level services such as local storage, real-time local data processing, embedded data mining, etc., proposing thus a Smart e-Health Gateway. By taking responsibility for handling some burdens of the sensor network and a remote healthcare center, a Smart e-Health Gateway can cope with many challenges in ubiquitous healthcare systems such as energy efficiency, scalability, and reliability issues. A successful implementation of Smart e-Health Gateways enables massive deployment of ubiquitous health monitoring systems especially in clinical environments. We also present a case study of a Smart e-Health Gateway called UTGATE where some of the discussed higher-level features have been implemented. Our proof-of-concept design demonstrates an IoT-based health monitoring system with enhanced overall system energy efficiency, performance, interoperability, security, and reliability.