In this paper we investigate the application of adaptive postfiltering for the enhancement of reverberant speech. The considered method is commonly used in Code Excited Linear Prediction (CELP) speech coding to lower the impact of quantization noise in the excitation signal and the spectral envelope. We show that the underlying additive noise model is accurate enough to enhance speech which is recorded in an enclosed space where the resulting early reflections are usually modeled as a convolutive distortion. By means of adaptive filtering, the amplitudes of the unwanted peaks in the excitation signal are attenuated and the signal components at the harmonic peaks are emphasized. Both, single- and multi-channel dereverberation algorithms are proposed having a moderate computational complexity. Experiments have shown that this approach is capable of reducing early reverberation and attenuate the ’distance-effect’ arising from room reflections.