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We propose a new blind multiuser detector for frequency-selective multipath channels. Our new linearly constrained minimum variance (LCMV) RAKE receiver is an enhancement to the previously published decorrelating RAKE (D-RAKE) receiver. Compared with the D-RAKE receiver, our LCMV-RAKE receiver processes full information of the desired user and is shown by simulations to provide lower bit-error rates. In some scenarios, our LCMV-RAKE receiver outperforms the minimum output energy detector by M.K. Tsatsanis and Z. Xu (see IEEE Trans. Sig. Processing, vol.46, p.3014-22, 1998). Adaptive implementation of our LCMV-RAKE receiver is demonstrated to be feasible.