We propose and demonstrate methods for suppressing the polarization dependence in the interrogation of birefringent fiber Bragg gratings. A wavelength-swept fiber laser with a polarized output was used as the light source. Two polarization-averaging methods, a depolarization scheme and a polarization scrambling scheme, were investigated and compared. The proposed techniques successfully stabilized the reflection spectrum of a birefringent grating regardless of the polarization state of the source laser and birefringence of the lead fiber. The results of this work eliminate one of the major practical difficulties in current fiber Bragg grating interrogation.