In the Internet of Things era, an increasing number of everyday objects are able to offer innovative services to the user. However, most of these devices provide only smartphone or web user interfaces. As a result, the interaction is disconnected from the physical world, decreasing the user experience and increasing the risk of user alienation from the physical world. We argue that tangible interaction can counteract this trend and this article discusses the potential benefits and the still open challenges of tangible interaction applied to the Internet of Things. After an analysis of open challenges for Human-Computer Interaction in IoT, we summarize current trends in tangible interaction and extrapolate eight tangible interaction properties that could be exploited for designing novel interactions with IoT objects. Through a systematic review of tangible interaction applied to IoT, we show what has been already explored in the systems that pioneered the field and the future explorations that still have to be conducted. In order to guide future work in this field, we propose a design card set for supporting the generation of tangible interfaces for IoT objects. The card set has been evaluated during a workshop with 21 people and the results are discussed.