There is a group of binary post-AGB stars that show conspicuous near infrared (NIR) excess, which is usually assumed to arise from hot dust in very compact possibly rotating disks. These stars are surrounded by significantly fainter nebulae than the standard, well studied protoplanetary and planetary nebulae (PPNe, PNe). We aim to identify and study extended rotating disks around these stars and shed light on the role of disks in the formation and shaping of planetary nebulae. We present high-sensitivity mm-wave observations of CO lines in 24 objects of this type. The resulting CO lines are compared with profiles expected to arise from rotating disks, from both theoretical and observational grounds. We derive simple formulae that allow us to determine the mass of the CO-emitting gas and estimate its extent. The reliability and uncertainty of the methods are also widely discussed. CO emission is detected in most observed sources, and the line profiles show that the emissions very probably come from disks in rotation. We derive typical values of the disk mass between 10^-3^ and 10^-2^M_{sun}_, about two orders of magnitude lower than the (total) masses of standard PPNe. The high-detection rate (upper limits being not very significant) clearly confirm that the NIR excess of these stars arises from compact disks in rotation, which are likely the inner parts of those found here. Low-velocity outflows are also found in about eight objects with moderate expansion velocities of ~10km/s, to be compared with the velocities of about 100km/s often found in standard PPNe. Except for two sources with complex profiles, the outflowing gas in our objects represents a minor nebular component. Our simple estimates of the typical disk sizes yields values ~0.5-1arcsec, which is between 5x10^15^ and 3x10^16^cm. Estimates of the linear momenta carried by the outflows, which can only be performed in a few well studied objects, also yield moderate values when compared to the linear momenta that can be released by the stellar radiation pressure (contrary, again, to the case of the very massive and fast bipolar outflows in $standard$ PPNe that are strongly overluminous). The mass and dynamics of nebulae around various classes of post-AGB stars differ very significantly, and we can expect the formation of PNe with very different properties.