We present a new method to generate steady and tunable, coherent, broadband terahertz radiation from a relativistic electron beam modulated by a femtosecond laser. We have demonstrated this in the electron storage ring at the Advanced Light Source. Interaction of an electron beam with a femtosecond laser pulse copropagating through a wiggler modulates the electron energies within a short slice of the electron bunch with about the same duration of the laser pulse. The bunch develops a longitudinal density perturbation due to the dispersion of electron trajectories, and the resulting hole emits short pulses of temporally and spatially coherent terahertz pulses synchronized to the laser. We present measurements of the intensity and spectra of these pulses. This technique allows tremendous flexibility in shaping the terahertz pulse by appropriate modulation of the laser pulse.