We investigate the relation between the projected morphology and the velocity dispersion of groups of galaxies using two recently compiled group catalogs, one based on the Two Micron All Sky Survey redshift survey (Crook et al.) and the other on the Sloan Digital Sky Survey Data Release 5 galaxy catalog (Tago et al.). We analyze a suitable subsample of groups from each catalog selected such that it minimizes possible systematic effects. We find that the velocity dispersion of groups is strongly correlated with the group-projected shape and size, with elongated and larger groups having a lower velocity dispersion. Such a correlation could be attributed to the dynamical evolution of groups, with groups in the initial stages of formation, before virialization is complete, having small velocity dispersion, a large size, and an elongated shape that reflects the anisotropic accretion of galaxies along filamentary structures. However, we show that the same sort of correlations could also be reproduced in prolatelike groups, irrespective of their dynamical state, if the net galaxy motion is preferentially along the group elongation, since then the groups oriented close to the line of sight will appear more spherical, will have a small projected size and high-velocity dispersion, while groups oriented close to the sky plane will appear larger in projection, more elongated, and will have smaller velocity dispersion. Although both factors must play a role in shaping the observed correlations, we attempt to disentangle them by performing tests that relate only to the dynamical evolution of groups (i.e., calculating the fraction of early-type galaxies in groups and the projected group compactness). Indeed we find a strong positive (negative) correlation between the group velocity dispersion (group-projected major axis) with the fraction of early-type galaxy members. We conclude that (1) the observed dependences of the group velocity dispersion on the group-projected size and shape, should be attributed mostly to the dynamical state of groups, and (2) groups of galaxies in the local universe do not constitute a family of objects in dynamical equilibrium, but rather a family of cosmic structures that are presently at various stages of their virialization process.