Three experiments were conducted to examine the effects of extraneous speech warnings (i.e., low-priority warnings initiated during high-priority tasks) on cognitive performance and whether organizing the auditory warnings into streams can attenuate any disruption. Experiment 1 demonstrated that a variety of speech warnings can be separated into perceptually distinct streams by allocating them to discrete spatial locations. Experiment 2 showed that increasing the rate of presentation of the warnings to promote streaming decreased clarity ratings but increased perceived urgency ratings. Experiment 3 demonstrated that the disruption to serial memory for navigational information by extraneous speech warnings could be attenuated by streaming. Results are interpreted in light of previous research, and practical implications for auditory warning design are discussed.