The pervasive need to safely share and store information between devices calls for the replacement of centralized trust architectures with decentralized ones. DLTs are seen as the most promising enabler of decentralized trust, but they still lack technological maturity, and their successful adoption depends on the understanding of the fundamental design trade-offs and their reflection in the actual technical design. This work focuses on the challenges and potential solutions for an effective integration of DLTs in the context of the Internet of Things (IoT). We first introduce the landscape of IoT applications and discuss the limitations and opportunities offered by DLTs. Then we review the technical challenges encountered in the integration of resource-constrained devices with distributed trust networks. We describe the common traits of lightweight synchronization protocols, and propose a novel classification rooted in the IoT perspective. We identify the need for receiving ledger information at the endpoint devices, implying a two-way data exchange that contrasts with the conventional uplink-oriented communication technologies intended for IoT systems.