How diverse are the outputs of large language models when diversity is desired? We examine the diversity of responses of several language models to questions with multiple possible answers, comparing them with human responses. Our findings suggest that models' responses are highly concentrated, reflecting narrow, mainstream outputs, in comparison to humans, whose responses exhibit a much longer-tail. We examine three simple and practical ways to increase output diversity: 1) increasing generation randomness via temperature sampling; 2) prompting models to answer from diverse perspectives using a single prompt; 3) aggregating outputs from several models. We find that these interventions, especially when combined, can substantially increase output diversity, although single-model outputs generally remain less diverse than the human baseline. We discuss potential implications of these findings for future work in AI policy and governance that wishes to preserve cultural diversity, an essential building block of a democratic social fabric.