In this letter, we propose a new multi-step maximum likelihood predictor with a finite impulse response (FIR) structure for discrete-time state-space signal models. This predictor is called a maximum likelihood FIR predictor (MLFP). The MLFP is linear with the most recent finite outputs and does not require a prior initial state information on a receding horizon. It is shown that the proposed MLFP possesses the unbiasedness property and the deadbeat property. Simulation study illustrates that the proposed MLFP is more robust against uncertainties and faster in convergence than the conventional multi-step Kalman predictor.