We present results from a study of the kinematic structure of star-forming galaxies at redshift z similar to 3 selected in the VIMOS VLT Deep Survey (VVDS), using integral field spectroscopy of rest-frame optical nebular emission lines, in combination with rest-frame UV spectroscopy, ground-based optical/near-IR and Spitzer photometry. We also constrain the underlying stellar populations to address the evolutionary status of these galaxies. We infer the kinematic properties of four galaxies: VVDS-20298666, VVDS-020297772, VVDS-20463884 and VVDS-20335183 with redshifts z = 3.2917, 3.2878, 3.2776 and 3.7062, respectively. While VVDS-20463884 presents an irregular velocity field with a peak in the local velocity dispersion of the galaxy shifted from the centre of the galaxy, VVDS-20298666 has a well-resolved gradient in velocity over a distance of similar to 4.5 kpc with a peak-to-peak amplitude of v = 91 km s(-1). We discovered that the nearby galaxy, VVDS-020297772 (which shows traces of active galactic nucleus activity), is in fact a companion at a similar redshift with a projected separation of 12 kpc. In contrast, the velocity field of VVDS-020335183 seems more consistent with a merger on a rotating disc. However, all of the objects have a high local velocity dispersion (sigma similar to 60-70 km s(-1)), which gives v/sigma less than or similar to 1. It is unlikely that these galaxies are a dynamically cold rotating disc of ionized gas.