The presence of a long secondary period (LSP) in the light curves of some local semiregular variables has been known for many years. Furthermore, the LSPs have recently been found in the light curves of approximately 25% of the semiregular variables in the LMC. They typically have a length of ~500-4000 days, some 5-15 times longer than the primary period. Binarity, pulsation, periodic dust ejection, and rotation have been suggested as the origin of the LSPs. Here we analyze echelle spectra of a group of local semiregular variables with LSPs (hereafter LSPVs) in order to try to distinguish between these suggestions. In general, we find that LSPVs do not have broader spectral features than semiregulars without a long secondary period (hereafter non-LSPVs). The general upper limit on the equatorial rotation velocity of 3 km s-1 rules out rotating spot and similar models. One LSPV, V Hya, does have broader spectral lines than similar carbon stars, but it is shown here that rotation alone is not a good model for explaining the broad lines. Mid-infrared colors of LSPs and non-LSPVs are similar and there are no LSPVs showing the large (60-25) μm IRAS color exhibited by some R Coronae Borealis (RCB) stars. Thus, there is no evidence for periodic dust ejection from LSPVs. Finally, we find that the LSPVs show larger radial velocity variations than non-LSPVs, which suggests that LSPs are caused either by binarity or by pulsation. A similar conclusion was derived by Hinkle and co-workers.