The emerging critique of alternative food networks (AFNs) points to several factors that could impede the participation of low-income, minority communities in the movement, namely, spatial and temporal constraints, and the lack of economic, cultural, and human capital. Based on a semi-experimental study that offers 6 weeks of free produce to 31 low-income African American households located in a New Orleans food desert, this article empirically examines the significance of the impeding factors identified by previous scholarship, through participant surveys before, during, and after the program. Our results suggest economic constraints are more influential in determining where the participants shop for food than spatial and temporal constraints, and the study participants exhibit high levels of human and cultural capital regarding the purchase and consumption of locally grown produce. We also find them undeterred by the market's predominantly White, middle-class cultural social space, which leads us to question the extent to which cultural exclusivity discourages their participation in AFNs. For all five factors we find that the constraints posed to accessing the local food market were not universal but varied among the participants. Finally, the study reveals some localized social constraints, fragmented social ties in particular, as a possible structural hurdle to engaging these residents in the alternative market in their neighborhood. Conclusions point to the need for a multi-dimensional and dynamic conceptualization of ''food access.''