The 30 Doradus (30 Dor) region of the Large Magellanic Cloud, also known as the Tarantula Nebula, is the nearest starburst region. It contains the richest population of massive stars in the Local Group and it is thus the best possible laboratory to investigate open questions in the formation and evolution of massive stars. Using ground based multi-object optical spectroscopy obtained in the framework of the VLT-FLAMES Tarantula Survey (VFTS), we aim to establish the (projected) rotational velocity distribution for a sample of 216 presumably single O-type stars in 30 Dor. The size of the sample is large enough to obtain statistically significant information and to search for variations among sub-populations - in terms of spectral type, luminosity class, and spatial location - in the field of view. We measured projected rotational velocities, Vrot, by means of a Fourier transform method and a profile fitting method applied on a set of isolated spectral lines. We also used an iterative deconvolution procedure to infer the probability density, P(Veq), of the equatorial rotational velocity, Veq. The distribution of Vrot shows a two-component structure: a peak around 80km/s and a high-velocity tail extending up to ~600km/s. This structure is also present in the inferred distribution P(Veq) with around 80% of the sample having 0<Veq<=300km/s and the other 20% distributed in the high-velocity region. The presence of the low-velocity peak is consistent with that found in other studies for late O- and early B-type stars. Most of the stars in our sample rotate with a rate less than 20% of their break-up velocity. For the bulk of the sample, mass-loss in a stellar wind and/or envelope expansion is not efficient enough to significantly spin down these stars within the first few Myr of evolution. If massive-star formation results in stars rotating at birth with a large fraction of their break-up velocities, an alternative braking mechanism, possibly magnetic fields, is thus required to explain the present day rotational properties of the O-type stars in 30 Dor. The presence of a sizeable population of fast rotators is compatible with recent population synthesis computations that investigate the influence of binary evolution on the rotation rate of massive stars. Despite the fact that we have excluded stars that show significant radial velocity variations, our sample may have remained contaminated by post-interaction binary products. The fact that the high-velocity tail may be preferentially (and perhaps even exclusively), populated by post-binary interaction products, has important implications for the evolutionary origin of systems that produce gamma-ray bursts.