In spiral galaxies in which the molecular phase dominates the ISM, the molecular gas as traced by CO emission will approximately obey the continuity equation on orbital timescales. The Tremaine-Weinberg method can then be used to determine the pattern speed of such galaxies. We have applied the method to single-dish CO maps of three nearby spirals, M51, M83, and NGC 6946, to obtain estimates of their pattern speeds: 38 +/- 7, 45 +/- 8, and 39 +/- 8 km s(-1) kpc(-1), respectively, and we compare these results to previous measurements. We also analyze the major sources of systematic errors in applying the Tremaine-Weinberg method to maps of CO emission.