In this paper, effects of birefringence on radio waves obliquely propagating though polar ice sheets are examined to facilitate interpretations of bistatic and side-looking radar data. A formalism applicable for arbitrary radar configurations is developed to predict the returned power from within and beneath the ice sheets that have arbitrary alignments of ice crystals (ice fabrics). We applied this formalism to a range of ice fabrics found in ice cores and assessed the effects of birefringence in terms of ray-path configurations, ice fabrics, and radar frequency. Predicted frequency dependence of the bed return power replicates prominent features observed at Greenland NGRIP ice-core site. Results show that birefringence in ice of 1 km or more thickness with strong (weak) fabric can reduce the power returned from the bed 2 dB or more at frequencies higher than 200 MHz (20 MHz) as compared to isotropic ice. This suggests that quantitative interpretation of the power returned from the bed requires careful assessment of birefringence almost everywhere over the ice sheets. Application of this formalism also suggests a radar-frequency range usable for attenuation measurements, possible effects of fabric on synthetic aperture radar processing, and a feasibility of remote sensing of ice fabric.