Recent significant advances in self-interference cancellation techniques pave the way for the deployment of full-duplex wireless transceivers capable of concurrent transmission and reception on the same channel. Despite the promise to theoretically double the spectrum efficiency, full-duplex prototyping in off-the-shelf chips of mobile devices is still in its infancy, mainly because of the challenges in mitigating self-interference to a tolerable level and the strict hardware constraints. In this article, we argue in favor of embedding full-duplex radios in onboard units of future vehicles. Unlike the majority of mobile devices, vehicular onboard units are good candidates to host complex FD transceivers because of their virtually unlimited power supply and processing capacity. Taking into account the effect of imperfect SI cancellation, we investigate the design implications of full-duplex devices at the higher-layer protocols of next-generation vehicular networks and highlight the benefits they could bring with respect to half-duplex devices in some representative use cases. Early results are also provided that give insight into the impact of self-interference cancellation on vehicle-to-roadside communications, and showcase the benefits of FD-enhanced medium access control protocols for vehicle-to-vehicle communications supporting crucial road safety applications.