This paper presents an efficient algorithm for high-quality speech capture in applications such as hands-free teleconferencing or voice recording by personal computers. We process the microphone signals by a subband adaptive filtering structure using a modulated complex lapped transform (MCLT), in which the subband filters are adapted to maximize the kurtosis of the linear prediction (LP) residual of the reconstructed speech. In this way, we attain good solutions to the problem of blind speech dereverberation. Experimental results with actual data, as well as with artificially difficult reverberant situations, show very good per-formance, both in terms of a significant reduction of the perceived reverberation, as well as improvement in spectral fidelity.