This paper investigates fundamental performance limits of medium access control (MAC) protocols for multi-hop sensor networks. A unique aspect of this study is the modeling of a fair-access criterion requiring that sensors have an equal rate of frame delivery to the base station. Tight upper bounds on network utilization and tight lower bounds on minimum time between samples are derived for fixed linear and grid topologies. The significance of these bounds is two-fold: First, they are universal, i.e., they hold for any MAC protocol. Second, they are provably tight, i.e., they can be achieved by a version of time division multiple access (TDMA) protocol that is self-clocking, and therefore does not require system-wide clock synchronization. The paper also examines the implication of the end-to-end performance bounds regarding the traffic rate and sensing time interval of individual sensors.