Requests by patients for providers of specific demographic backgrounds pose an ongoing challenge for hospitals, policymakers, and ethicists. These requests may stem from a wide variety of motivations; some may be consistent with broader societal values, although many others may reflect prejudices inconsistent with justice, equity, and decency. This paper proposes a taxonomy designed to assist healthcare institutions in addressing such cases in a consistent and equitable manner. The paper then reviews a range of ethical and logistical challenges raised by such requests and proposed guidance to consider when reviewing and responding to them.