We report the selection and spectroscopic confirmation of 129 new late-type (K3-M6) members of the Tuc-Hor moving group, a nearby (~40 pc), young (~40 Myr) population of comoving stars. We also report observations for 13/17 known Tuc-Hor members in this spectral type range, and that 62 additional candidates are likely to be unassociated field stars; the confirmation frequency for new candidates is therefore 129/191 = 67%. We have used RVs, Halpha emission, and Li6708 absorption to distinguish contaminants and bona fide members. Our expanded census of Tuc-Hor increases the known population by a factor of ~3 in total and by a factor of ~8 for members with SpT>K3, but even so, the K-M dwarf population of Tuc-Hor is still markedly incomplete. The spatial distribution of members appears to trace a 2D sheet, with a broad distribution in X and Y, but a very narrow distribution (+/-5 pc) in Z. The corresponding velocity distribution is very small, with a scatter of +/-1.1 km/s about the mean UVW velocity. We also show that the isochronal age (20--30 Myr) and the lithium depletion age (40 Myr) disagree, following a trend seen in other PMS populations. The Halpha emission follows a trend of increasing EW with later SpT, as seen for young clusters. We find that members have been depleted of lithium for spectral types of K7.0-M4.5. Finally, our purely kinematic and color-magnitude selection procedure allows us to test the efficiency and completeness for activity-based selection of young stars. We find that 60% of K-M dwarfs in Tuc-Hor do not have ROSAT counterparts and would be omitted in Xray selected samples. GALEX UV-selected samples using a previously suggested criterion for youth achieve completeness of 77% and purity of 78%. We suggest new selection criteria that yield >95% completeness for ~40 Myr populations.(Abridged)
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