Context. 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 on the formation and evolution of massive stars. Aims: 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 sample is large enough to obtain statistically significant information and to search for variations among subpopulations - in terms of spectral type, luminosity class, and spatial location - in the field of view. Methods: We measured projected rotational velocities, νesini, by means of a Fourier transform method and a profile fitting method applied to a set of isolated spectral lines. We also used an iterative deconvolution procedure to infer the probability density, P(νe), of the equatorial rotational velocity, νe. Results: The distribution of νesini shows a two-component structure: a peak around 80 kms-1 and a high-velocity tail extending up to ~600 kms-1. This structure is also present in the inferred distribution P(νe) with around 80% of the sample having 0 <νe ≤ 300 kms-1 and the other 20% distributed in the high-velocity region. The presence of the low-velocity peak is consistent with what has been found in other studies for late O- and early B-type stars. Conclusions: 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 portion 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. Even though we have excluded stars that show significant radial velocity variations, our sample may have remained contaminated by post-interaction binary products. That the high-velocity tail may be populated primarily (and perhaps exclusively) by post-binary interaction products has important implications for the evolutionary origin of systems that produce gamma-ray bursts. Based on observations collected at the European Southern Observatory under program ID 182.D-0222.Full Tables 3 and 4 are only available at the CDS via anonymous ftp to http://cdsarc.u-strasbg.fr (ftp://130.79.128.5) or via http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/A+A/560/A29