The isolated binary evolution model for merging neutron stars (NSs) involves processes such as mass transfer, common-envelope evolution, and natal kicks, all of which are poorly understood. Also, the predicted NS-NS merger rates are typically lower than the rates inferred from the LIGO GW170817 event. Here, we investigate merger rates of NS and black hole-NS binaries in hierarchical triple-star systems. In such systems, the tertiary can induce Lidov-Kozai (LK) oscillations in the inner binary, accelerating its coalescence and potentially enhancing compact object merger rates. However, because compact objects originate from massive stars, the prior evolution should also be taken into account. Natal kicks, in particular, could significantly reduce the rates by unbinding the tertiary before it can affect the inner binary through LK evolution. We carry out simulations of massive triples, taking into account stellar evolution starting from the main sequence, secular and tidal evolution, and the effects of supernovae. For large NS birth kicks (sigma(k) = 265 km s(-1)), we find that the triple NS-NS merger rate (several hundred Gpc(-3) yr(-1)) is lower by a factor of similar to 2-3 than the binary rate, but for no kicks (sigma(k) = 0 km s(-1)), the triple rate (several thousand Gpc(-3) yr(-1)) is comparable to the binary rate. Our results indicate that a significant fraction of NS-NS mergers could originate from triples if a substantial portion of the NS population is born with low kick velocities, as indicated by other work. However, uncertainties and open questions remain because of our simplifying assumption of dynamical decoupling after inner binary interaction has been triggered.