Recent angular diameter measurements for Mira variables suggest that the radii of these stars are very large and consistent with pulsation in the first-overtone mode rather than the fundamental mode. On the other hand, non-linear pulsation models of Mira variables suggest that the observed pulsation velocity amplitudes can only be achieved during fundamental-mode pulsation, at least for stellar masses <~2.0M solar. Here, we present some new observations of long-period variables (LPVs) in the LMC which show that the LPVs lie on two (K, logP) sequences, one sequence being the well-known Mira sequence and the other being a sequence parallel to the Mira sequence but separated from it by DeltalogP ~ 0.35. The LPVs on the Mira sequence have a wide range of amplitudes (0.1 < DeltaI < 3) while those on the second sequence have relatively small amplitudes (DeltaI < 0.5). The previously known LPVs of large amplitude (DeltaI > 0.5) in the LMC lie almost always on the Mira sequence. Theoretical models of LPVs predict a ratio of fundamental to first-or second-overtone period of DeltalogP~ 0.3-0.4, and overtone pulsators are expected to have smaller limiting amplitudes than fundamental-mode pulsators. Hence the above observations can be easily understood if the LPVs on the Mira sequence are fundamental-mode pulsators while LPVs on the second sequence are overtone pulsators. A second test of the pulsation mode is obtained by computing pulsation periods for model stars on the LMC old giant branch and comparing these periods with those of observed Mira variables. Once again, the fundamental-mode pulsators have periods consistent with those seen in the LMC Miras while the overtone periods are too short. The above results strongly suggest that Mira variables are fundamental-mode pulsators.