In a vehicular ad hoc network (VANET), vehicles attempt to improve safety by emitting safety messages that others receive to increase their awareness of one another. Yet, the conflicting data delivery requirements of safety applications can potentially jeopardize safety itself. While many information dissemination techniques have been proposed to improve data delivery in a VANET, such as flooding and geocasting, evaluations do not focus on measuring safety efficiency in terms of safety application requirements and often fail to include mobility considerations, such as time to contact (TTC). To improve safety in a VANET, we propose SafeRelay, a flooding-based message dissemination technique that relays safety messages within a geographically addressed forwarding zone (FZ). We evaluate different forwarding zone sizes in terms of several metrics, including safety awareness probability, which combines both communications and mobility performance. Simulation results of an urban downtown scenario show that SafeRelay can significantly improve safety awareness using moderately-sized nearby forwarding zones.