Aims: The incidence of multiplicity in cool, luminous massive stars is relatively unknown compared to their hotter counterparts. In this work we present radial velocity (RV) measurements and investigate the multiplicity properties of red supergiants (RSGs) in the 30 Doradus region of the Large Magellanic Cloud using multi-epoch visible spectroscopy from the VLT-FLAMES Tarantula Survey. Methods: Exploiting the high density of absorption features in visible spectra of cool stars, we used a novel slicing technique to estimate RVs of 17 candidate RSGs in 30 Doradus from cross-correlation of the observations with model spectra. Results: We provide absolute RV measurements (precise to better than ±1 km s-1) for our sample and estimate line-of-sight velocities for the Hodge 301 and SL 639 clusters, which agree well with those of hot stars in the same clusters. By combining results for the RSGs with those for nearby B-type stars, we estimate systemic velocities and line-of-sight velocity dispersions for the two clusters, obtaining estimates for their dynamical masses of log(Mdyn/M⊙) = 3.8 ± 0.3 for Hodge 301, and an upper limit of log(Mdyn/M⊙) < 3.1 ± 0.8 for SL 639, assuming virial equilibrium. Analysis of the multi-epoch data reveals one RV variable, potential binary candidate (VFTS 744), which is likely a semi-regular variable asymptotic giant branch star. Calculations of semi-amplitude velocities for a range of RSGs in model binary systems and literature examples of binary RSGs were used to guide our RV variability criteria. We estimate an upper limit on the observed binary fraction for our sample of 0.3; for this sample we are sensitive to maximum periods for individual objects in the range 1-10 000 days and mass ratios above 0.3 depending on the data quality. From simulations of RV measurements from binary systems given the current data, we conclude that systems within the parameter range q > 0.3, log P [days] < 3.5 would be detected by our variability criteria at the 90% confidence level. The intrinsic binary fraction, accounting for observational biases, is estimated using simulations of binary systems with an empirically defined distribution of parameters in which orbital periods are uniformly distributed in the 3.3 < log P [days] < 4.3 range. A range of intrinsic binary fractions are considered; a binary fraction of 0.3 is found to best reproduce the observed data. Conclusions: We demonstrate that RSGs are effective extragalactic kinematic tracers by estimating the kinematic properties, including the dynamical masses of two LMC young massive clusters. In the context of binary evolution models, we conclude that the large majority of our sample consists of effectively single stars that are either currently single or in long-period systems. Further observations at greater spectral resolution or over a longer baseline, or both, are required to search for such systems. Full Table A.1 is 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/624/A129Based on observations collected at the European Organisation for Astronomical Research in the Southern Hemisphere under ESO programme 182.D-0222.