We show that the use of a dispersion oscillating fiber in passive cavities significantly extends the modulational instability to novel high-frequency bands, which also destabilizes the branches of the steady response that are stable with homogeneous dispersion. By means of Floquet theory, we obtain the exact explicit expression for the sideband gain, and a simple analytical estimate for the frequencies of the maximum gain. Numerical simulations show that stable stationary trains of pulses can be excited in the cavity.