Many Internet of Things (IoT) networks comprise tiny devices with limited processing power and tight energy restrictions. These limitations make it difficult to use established security mechanisms protecting the devices against malicious attacks. This holds particularly for Intrusion Detection Systems (IDS) that help to detect various net-based attacks but often demand significant network and computing resources. In this article, we design and evaluate some IDS mechanisms for IoT Networks that are suited to small devices. They use a trust management mechanism that allows devices to manage reputation information about their neighbors. This mechanism makes it possible to single out maliciously behaving units in a processing and energy-friendly way. The approach is explained in the context of the healthcare domain.