We have studied two-body charmless decays of the B meson into the final states rho(0)rho(0), K(*0)rho(0), K(*0)K(*0), K(*0)K(*0), K(*+)rho(0), K(*+)K(*0), and K(*+)K(*-) using only decay modes with charged daughter particles. Using 9.7x10(6) BB pairs collected with the CLEO detector, we place 90% confidence level upper limits on the branching fractions (1.4-14.1)x10(-5), depending on final state and polarization.