Instruction-tuned Large Language Models (LLMs) can perform a wide range of tasks given natural language instructions to do so, but they are sensitive to how such instructions are phrased. This issue is especially concerning in healthcare, as clinicians are unlikely to be experienced prompt engineers and the potential consequences of inaccurate outputs are heightened in this domain. This raises a practical question: How robust are instruction-tuned LLMs to natural variations in the instructions provided for clinical NLP tasks? We collect prompts from medical doctors across a range of tasks and quantify the sensitivity of seven LLMs—some general, others specialized—to natural (i.e., non-adversarial) instruction phrasings. We find that performance varies substantially across all models, and that—perhaps surprisingly—domain-specific models explicitly trained on clinical data are especially brittle, compared to their general domain counterparts. Further, arbitrary phrasing differences can affect fairness, e.g., valid but distinct instructions for mortality prediction yield a range both in overall performance, and in terms of differences between demographic groups.