Accretion-powered millisecond X-ray pulsations have been detected in only 13 of more than 130 known neutron stars in low-mass X-ray binary systems (LMXBs). Pulsations have been detected only intermittently in three of these accretion-powered millisecond X-ray pulsars. We show that the failure to detect accretion-powered pulsations from most neutron-star LMXBs, and the intermittent detection of pulsations in a few, can both be explained if the emitting regions of accreting neutron stars with millisecond spin periods are typically close to their spin poles and move in response to changes in the accretion flow. This nearly-aligned moving spot model is consistent with the small observed amplitudes and nearly sinusoidal waveforms of most accreting millisecond X-ray pulsars, and the large, rapid phase variations of several.Fast X-ray timing and spectroscopy at extreme count rates