This paper investigates the performance of a new technique for speech enhancement which combines Linear Predictive (LP) spectrum-based perceptual filtering to the recordings obtained from an Acoustic Vector Sensor (AVS). The technique takes advantage of the directional polar responses of the AVS to obtain a significantly more accurate representation of the LP spectrum of a target speech signal in the presence of noise when compared to single channel, omni-directional recordings. Comparisons between the speech quality obtained from the proposed technique and existing beamforming-based speech enhancement techniques for the AVS are made through Perceptual Evaluation of Speech Quality (PESQ) tests and Mean Opinion Score (MOS) listening tests. Results show significant improvements in PESQ and MOS scores of 0.2 and 1.6, respectively, for the proposed enhancement technique. Being based on a miniature microphone array, the approach is particular suitable for hands free communication applications in mobile telephony.