We present 12CO J=1-->0 observations of ultraluminous infrared galaxies (ULIRGs) obtained using the Redshift Search Receiver (RSR) on the 14-m telescope of the Five College Radio Astronomy Observatory. The RSR is a novel, dual-beam, dual-polarization receiver equipped with an ultra-wideband spectrometer backend that is being built as a facility receiver for the Large Millimeter Telescope. Our sample consists of 29 ULIRGs in the redshift range of 0.04-0.11, including 10 objects with no prior 12CO measurements. We have detected 27 systems (a detection rate of 93%), including 9 ULIRGs that are detected in CO for the first time. Our study has increased the number of local ULIRGs with CO measurements by ~15%. The CO line luminosity L'_CO, correlates well with far-infrared luminosity L_FIR, following the general trend of other local ULIRGs. However, compared to previous surveys we probe deeper into the low CO luminosity end of ULIRG population as a single study by including a number of CO faint objects in the sample. As a result, we find 1) a smoother transition between the ULIRG population and local QSOs in L_FIR-L'_CO ("star formation efficiency") space, and 2) a broader range of L_FIR/L'_CO flux ratio (~60--1000 L_sun/[K km/s/pc^2]) than previously reported. In our new survey, we also have found a small number of ULIRGs with extreme L_FIR/L'_CO which had been known to be rare. The mid-IR color and radio-excess of 56 local ULIRGs as a function of FIR-to-CO flux ratio is examined and compared with those of spirals/starburst galaxies and low-z QSOs. In this paper, using a large sample of local ULIRGs we explore the origin of their current power source and potential evolution to QSOs.