We searched for molecular gas in a heterogeneous sample of five radio-
loud galaxies (three of which are inferred to be in cooling flow clust
ers) using the Swedish-ESO Submillimeter Telescope. We do not detect C
O in emission in any of the cluster sources at a 3 sigma level of typi
cally 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 s
patial covering factor of order unity and a total mass of M approximat
ely 10(12) M.. Our limits are inconsistent with these column densities
and spatial covering factor unless the molecular gas is very cold (ki
netic temperature close to 2.7 K) or there only a few clouds along eac
h line of sight. We calculate the heating by X-rays from the intraclus
ter medium and derive constraints on the equilibrium temperatures of t
he molecular clouds. We estimate minimum temperatures in the range app
roximately 20-30 K. These calculations suggest that it is not possible
to cool the clouds sufficiently to explain the nondetections of molec
ular gas as a result of low temperature (kinetic temperature close to
2.7 K). We find that clouds of atomic and molecular hydrogen require s
trict fine-tuning of parameter space in order to satisfy the requireme
nts for the large column densities N(H) approximately 10(21) cm-2, uni
t covering factor, and a small number of clouds,along the line of sigh
t. The combination of these constraints with the additional requiremen
t that the optical depth in H I be very large is inconsistent with the
clouds being atomic. Clouds of molecular hydrogen are not currently r
uled out, but the range of parameter space is shrinking. Currently the
only way molecular gas can be responsible for the X-ray absorption an
d still be consistent with our observations is if (1) there is of orde
r 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 N
GC 4696 which shows a diffuse radio morphology comparable to that of t
he dust lane and emission line complex and suggest this object is a me
mber 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 class
ical double radio galaxy which is rich in extended optical emission li
ne gas. The estimated molecular gas mass is M(mol) approximately 3 x 1
0(9) M. and is much larger than that of the ionized component detected
in Halpha suggesting that the emission-line nebula is radiation bound
ed.