We use the ROSAT All Sky Survey to compute X-ray luminosities for an optically selected sample of 32 galaxy groups. The groups were identified by an objective algorithm as three-dimensional density enhancements in the magnitude-limited Center for Astrophysics Redshift Survey. The sample of groups includes five poor Abell clusters. We use the optical galaxy positions to limit the sky area in which we search for the faint, extended X-ray emission from the groups. Nine of the groups have significant diffuse X-ray emission, with 0.5-2.0 keV luminosities in the range 0.12-1.6 h(-2) x 10(43) ergs s(-1), where we use h = H-0/(50 km s(-1) Mpc(-1)). Among these nine groups, only three are Abell clusters. For the remaining 23 systems we compute 3 sigma upper limits. The relationship between the X-ray luminosities and the radial velocity dispersions differs significantly from that for rich clusters. We find L-X(group) proportional to sigma(1.56+/-0.25), whereas L-X(cluster) similar to sigma(4). This shallow slope is consistent with models in which intragroup gas concentrated around individual galaxies makes a substantial contribution to the extended X-ray emission. Previous work suggests that in this case L-X proportional to sigma(1.6) The X-ray/optical luminosity relationship for these systems is L-X proportional to L-B(1.06+/-0.11), physically consistent B with the shallow L-X - sigma relation. There is some correlation (Kendall's tau = 0.4, P = 0.1) between the fraction of member ellipticals and L-X.