We present the first demonstration of an inertially sensitive atomic interferometer based on a continuous, rather than pulsed, atomic beam at sub-Doppler temperatures in three dimensions. We demonstrate 30% fringe contrast in continuous, inertially sensitive interference fringes at interrogation time T = 6.7 ms and a short-term phase measurement noise of 530 µrad/ √ Hz as inferred from interference measurements. Atoms are delivered to the interferometer by a cold-rubidium source that produces a high flux of atoms at temperature ≤ 15 µK in three dimensions while reducing near-resonance fluorescence in the downstream path of the atoms. We describe the optimization of the interrogating Raman beams to achieve high contrast, and validate interferometer operation through comparison with measurements by commercial accelerometers. We further provide a demonstration of zero-dead-time phase-shear readout of atom interferometer phase, achieving a measurement rate up to 160 Hz. This demonstration lays the groundwork for future gyroscope/accelerometer sensors that measure continuously, with both high bandwidth and high sensitivity, and on dynamic platforms.