Approximately 25%-30% of pulsating asymptotic giant branch (AGB) stars show long secondary periods (LSPs) of typical length ~400-1500 days, roughly 10 times longer than the primary period of pulsation. Here we seek an explanation for the LSPs. We describe the spectral variations over a 4 yr interval for three of these stars. The radial velocity is found to vary during the LSP with full amplitude of ~5 km s-1, a result similar to that found by Hinkle and coworkers for six other variables of this type. Variations in the Hα and Na D line profiles throughout the LSP suggest that chromospheric activity and mass loss vary with the period of the LSP. Possible explanations for the photometric and radial velocity variations include eccentric motion of an orbiting companion of mass ~0.1 Msolar, radial and nonradial pulsation, rotation of an ellipsoidal-shaped red giant, episodic dust ejection, and star spot cycles. We discuss each of these models and show that they all have problems. The most likely explanation is that the LSPs result from a low degree g+ mode confined to the outer radiative layers of the red giant, combined with large-scale star spot activity to give the observed chromosphere and the irregularity of the light curve. We suggest that these stars may be the precursors of asymmetric planetary nebulae.