Using the IRAM 30m telescope we have obtained seven new, deep CO J(1-0) and J(2-1) observations of low surface brightness (LSB) galaxies. Five of the galaxies have no CO detected to extremely low limits (0.1-0.4 K km/s at J(1-0)), while two of the galaxies, UGC 01922 and UGC 12289, have clear detections in both line transitions. When these observations are combined with all previous CO observations taken of LSB systems, we compile a total of 34 observations, in which only 3 galaxies have had detections of their molecular gas. Comparing the LSB galaxies with and without CO detections to a sample of high surface brightness (HSB) galaxies with CO observations indicates that it is primarily the low density of baryonic matter within LSB galaxies which is causing their low CO fluxes. Finally, we note that one of the massive LSB galaxies studied in this project, UGC 06968 (a Malin-1 `cousin'), has upper limits placed on both M_H2 and M_H2/M_HI which are 10-20 times lower than the lowest values found for any galaxy (LSB or HSB) with similar global properties. This may be due to an extremely low temperature and metallicity within UGC 06968, or simply due to the CO distribution within the galaxy being too diffuse to be detected by the IRAM beam.