This Letter reports on measurements of the longitudinal
polarization fractions in the decays B_d -> J/psi K^*0 and B_s -> J psi phi
using data collected with the Collider Detector at Fermilab.  B_d mesons are
reconstructed through the decay chain B_d -> J/psi K^*0, J/psi -> mu^+mu^-,
K^*0 -> K^+pi^-.  A sample of 65 +/- 10 B_d events is used to obtain a
longitudinal polarization fraction of Gamma_L/Gamma = 0.65 +/- 0.10 (stat) +/-
0.04 (sys).  B_s mesons are reconstructed through the decay chain B_s -> J/psi
phi, J/psi -> mu^+mu^-, phi -> K^+K^-.  A sample of 19 +/- 5 B_s events is
used to obtain the result Gamma_L/Gamma = 0.56 +/- 0.21 (stat)^+0.02_ -0.04
(sys).