The Transiting Exoplanet Survey Satellite (TESS) searches for planets transiting bright and nearby stars using high-cadence, large-scale photometric observations. Full frame images provided by the TESS mission include a large number of serendipitously observed main-belt asteroids (MBAs). Due to the cadence of the published full frame images, we are sensitive to periods as long as of order tens of days, a region of phase space that is generally not accessible through traditional observing. This work represents a much less biased measurement of the period distribution in this period range. We have derived rotation periods for 300 MBAs and have partial lightcurves for a further 7277 asteroids, including 43 with periods of P>100hr; this large number of slow rotators is predicted by theory. Of these slow rotators we find none requiring significant internal strength to resist rotational reshaping. We find our derived rotation periods to be in excellent agreement with results in the Lightcurve Database for the 55 targets that overlap. Over the nominal two-year lifetime of the mission, we expect the detection of around 85,000 unique asteroids with rotation period solutions for around 6000 asteroids. We project that the systematic analysis of the entire TESS data set will increase the number of known slow-rotating asteroids (period >100hr) by a factor of 10. Comparing our new period determinations with previous measurements in the literature, we find that the rotation period of asteroid (2320) Blarney has decreased by at least 20% over the past decade, potentially due to surface activity or subcatastrophic collisions.