Context. Powerful radio-AGN are hosted by massive elliptical galaxies that are usually very poor in molecular gas. Nevertheless, gas is needed at their very center to feed the nuclear activity. Aims. We study the molecular gas properties (i.e., mass, kinematics, distribution, origin) of these objects, and compare them with results for other known samples. Methods. At the IRAM-30m telescope, we performed a survey of the CO(1-0) and CO(2-1) emission from the most powerful radio galaxies of the Local Universe, selected only on the basis of their radio continuum fluxes. Results. The main result of our survey is that the molecular gas content of these galaxies is very low compared to spiral or FIR-selected galaxies. The median value of the molecular gas mass, including detections and upper limits, is 2.2 × 10 8 M . When separated into FR-I and FR-II types, a difference in their H 2 masses is found. The median value of FR-I galaxies is about 1.9 × 10 8 M and higher for FR-II galaxies, at about 4.5 × 10 8 M . Which is probably entirely because of a Malmquist bias. Our results contrast with those of previous surveys, whose targets were mainly selected by means of their FIR emission, implying that we measure higher observed masses of molecular gas. Moreover, the shape of CO spectra suggest that a central molecular gas disk exists in 30% of these radio galaxies, a lower rate than in other active galaxy samples. Conclusions. We find a low level of molecular gas in our sample of radio-selected AGNs, indicating that galaxies do not need much molecular gas to host an AGN. The presence of a molecular gas disk in some galaxies and the wide range of molecular gas masses may be indicative of different origins for the gas, which we can not exclude at present (e.g., minor/major mergers, stellar mass loss, or accretion).