Approximately one third of non-hospitalized coronavirus disease of 2019 (COVID-19) patients report chronic symptoms after recovering from the acute stage of severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) infection. Some of the most persistent and common complaints of this post-acute COVID-19 syndrome (PACS) are cognitive in nature, described subjectively as "brain fog" and also objectively measured as deficits in executive function, working memory, attention, and processing speed. The mechanisms of these chronic cognitive sequelae are currently not understood. SARS-CoV-2 inflicts damage to cerebral blood vessels and the intestinal wall by binding to angiotensin-converting enzyme 2 (ACE2) receptors and also by evoking production of high levels of systemic cytokines, compromising the brain's neurovascular unit, degrading the intestinal barrier, and potentially increasing the permeability of both to harmful substances. Such substances are hypothesized to be produced in the gut by pathogenic microbiota that, given the profound effects COVID-19 has on the gastrointestinal system, may fourish as a result of intestinal post-COVID-19 dysbiosis. COVID-19 may therefore create a scenario in which neurotoxic and neuroinflammatory substances readily proliferate from the gut lumen and encounter a weakened neurovascular unit, gaining access to the brain and subsequently producing cognitive deficits. Here, we review this proposed PACS pathogenesis along the gut-brain axis, while also identifying specific methodologies that are currently available to experimentally measure each individual component of the model.