Estimating observability matrices or state sequences is the central component of existing subspace identification methods. In this paper a different approach, in which Markov parameters are first estimated under general input excitation, is proposed. The prominent difference of this approach is that a three-block arrangement of data matrices is used. It is shown that one advantage of this approach over other subspace algorithms is that several unbiased estimating procedures can be carried out. One immediate application is to obtain balanced or nearly balanced models directly from the estimated Markov parameters. Another application is that with the estimated Markov parameters, consistently initialized Kalman filter state sequences can be obtained, from which the system matrices can be easily determined without bias. Performance of the proposed algorithms is investigated in two case studies which are based on real data taken from two industrial systems. The algorithms developed in this paper have been implemented and are publicly available.