This paper presents a new statistical approach to the blind estimation of linear multiple-input multiple-output (MIMO) channels with finite impulse response. A matrix pencil is constructed from a set of fourth-order cumulant matrices of the channel output signals. The MIMO channel impulse responses can then be efficiently estimated from the generalized eigendecomposition of this cumulant matrix pencil. Random weighting is applied in the matrix pencil construction to improve the reliability of the algorithm. The proposed new method requires a relaxed channel identifiability condition and is robust in the sense that it does not require the exact knowledge of the MIMO channel order.