We use multifiber spectroscopy of 12 poor groups of galaxies to address (1) whether the groups are bound systems or chance projections of galaxies along the line of sight; (2) why the members of each group have not already merged to form a single galaxy, despite the groups' high galaxy densities, short crossing times, and likely environments for galaxy-galaxy mergers; and (3) how galaxies might evolve in these groups, where the collisional effects of the intragroup gas and the tidal influences of the global potential are weaker than in rich clusters. Each of the 12 groups has fewer than about five cataloged members in the literature. Our sample consists of 1002 galaxy velocities, 280 of which are group members. The groups have mean recessional velocities between 1600 and 7600km/s. Nine groups, including three Hickson compact groups, have the extended X-ray emission characteristic of an intragroup medium (see Paper II, 1998ApJ...496...73M).