This paper introduces a novel paradigm for designing the physical and medium access control (MAC) layers of mobile ad hoc or peer-to-peer networks formed by half-duplex radios. A node equipped with such a radio cannot simultaneously transmit and receive useful signals at the same frequency. Unlike in conventional designs, where a node's transmission frames are scheduled away from its reception, each node transmits its signal through a randomly generated on-off duplex mask (or signature) over every frame interval, and receive a signal through each of its own off-slots. This is called rapid on-off-division duplex (RODD). Over the period of a single frame, every node can transmit a message to some or all of its peers, and may simultaneously receive a message from each peer. Thus RODD achieves virtual full-duplex communication using half-duplex radios and can simplify the design of higher layers of a network protocol stack significantly. The throughput of RODD is evaluated under some general settings, which is significantly larger than that of ALOHA. RODD is especially efficient in case the dominant traffic is simultaneous broadcast from nodes to their one-hop peers, such as in spontaneous wireless social networks, emergency situations or on battlefield. Important design issues of peer discovery, distribution of on-off signatures, synchronization and error-control coding are also addressed.