Sufficiently strong security and privacy mechanisms are prerequisite to amass the promising benefits of the IoT technology and to incorporate this technology into our daily lives. This paper introduces a novel approach to privacy in networks, an approach which is especially well matched with the IoT characteristics. Our general approach is based on continually changing the identifying attributes of IoT nodes. In particular, the scheme proposed in this work is based on changing the IoT nodes' IP addresses, and because the changing patterns of the IP addresses appear random to a non-intended observer, an adversary is unable to identify the source or destination of a particular transmission. Thus, packets that carry information generated by a particular node cannot be linked together. The scheme offers additional security benefits, including DoS mitigation, is relatively easy to implement, and requires no changes to the existing networking infrastructure. We discuss the details of the implementation of the scheme and evaluate its performance.