We present the results of a study of the star formation rates (SFRs) of a sample of disk galaxies in nearby compact groups compared with the SFRs of a sample of field galaxies. For this purpose, Hα luminosities and equivalent widths were derived for the galaxies of our sample. A direct comparison of the equivalent widths and Hα luminosities, normalized to the B luminosities and estimated area of the galaxies of both samples, yields the result that the median values of these quantities are almost identical for both samples, although the distributions for the compact-group sample are broader around the mean value than was found in the field galaxy sample. This result can be explained by assuming that although interactions between galaxies in compact groups can alter the SFRs, the median value of the normalized SFRs is preserved, being almost indistinguishable from the corresponding value for field galaxies. Measuring the global L_Hα/L_B of the groups, including early-type galaxies, we find that most of the groups that show the highest level of L_Hα/L_B with respect to a set of synthetic groups built out of field galaxies show tidal features in at least one of their members. Finally, we have explored the relationship between the ratio L_Hα/L_B and several relevant dynamical parameters of the groups: velocity dispersion, crossing time, radius, and the mass-to-luminosity ratio, finding no clear correlation. This suggests that the exact dynamical state of a group does not control the SFR of the group as a whole. Our results are compatible with a scenario for compact groups of galaxies in which the dark matter of the group is arranged in a common halo, therefore preventing a fast collapse of the galaxies.