We report high-sensitivity millimeter-wave CO observations of recently discovered bipolar planetary nebulae (PNe). Three objects (BV 5-1, K 3-94 and K 3-24) have been detected, and one of them (BV 5-1) is resolved by the ~ 10{''} telescope beam. The envelopes of the three newly detected objects display values of the molecular to ionized mass ratio of ~ 0.2, and sizes of ~ 0.1 pc. This indicates that these PNe are rather evolved, with K 3-24 being the youngest of the three ( ~ 103 years) according to its CO line profile and kinematic ages. In BV 5-1, the molecular gas appears to be distributed in an irregular disk or ring surrounding the central star and perpendicular to the bipolar nebula. This is similar to the situation found in other better studied bipolar PNe (e.g. NGC 2346, M 2-9, KjPn8), and thus seems to be the rule in this class of objects. This suggests a common mechanism for the rings and the bipolar flows that shape the optical nebula.