We searched for molecular gas in a heterogeneous sample of five radio-loud galaxies (three of which are inferred to be in cooling flow clusters) using the Swedish-ESO Submillimeter Telescope. We do not detect CO in emission in any of the cluster sources at a 3 sigma level of typically 15 mK. White et al. (1991) have suggested that the apparent low-energy X-ray absorption toward cooling flow clusters is due to column densities of N(H) approximately 10(21) cm-2 in these clusters with a spatial covering factor of order unity and a total mass of M approximately 10(12) M.. Our limits are inconsistent with these column densities and spatial covering factor unless the molecular gas is very cold (kinetic temperature close to 2.7 K) or there only a few clouds along each line of sight. We calculate the heating by X-rays from the intracluster medium and derive constraints on the equilibrium temperatures of the molecular clouds. We estimate minimum temperatures in the range approximately 20-30 K. These calculations suggest that it is not possible to cool the clouds sufficiently to explain the nondetections of molecular gas as a result of low temperature (kinetic temperature close to 2.7 K). We find that clouds of atomic and molecular hydrogen require strict fine-tuning of parameter space in order to satisfy the requirements for the large column densities N(H) approximately 10(21) cm-2, unit covering factor, and a small number of clouds,along the line of sight. The combination of these constraints with the additional requirement that the optical depth in H I be very large is inconsistent with the clouds being atomic. Clouds of molecular hydrogen are not currently ruled out, but the range of parameter space is shrinking. Currently the only way molecular gas can be responsible for the X-ray absorption and still be consistent with our observations is if (1) there is of order one cloud along the line of sight and (2) the optical depth in (CO)-C-12 1 --> 0 is less than 10. In addition, we present a VLA image of NGC 4696 which shows a diffuse radio morphology comparable to that of the dust lane and emission line complex and suggest this object is a member of the class of ''amorphous cooling flow radio sources.'' The (CO)-C-12 1 --> 0 line is detected in emission in PKS 0634 - 206, a classical double radio galaxy which is rich in extended optical emission line gas. The estimated molecular gas mass is M(mol) approximately 3 x 10(9) M. and is much larger than that of the ionized component detected in Halpha suggesting that the emission-line nebula is radiation bounded.