In this work, we present a detailed study of the rotational properties of magnetized and self-gravitating dense molecular cloud (MC) cores formed in a set of two very high resolution three-dimensional (3D) MC simulations with decaying turbulence. The simulations have been performed using the adaptative mesh refinement code RAMSES with an effective resolution of 40963 grid cells. One simulation represents a mildly magnetically supercritical cloud and the other a strongly magnetically supercritical cloud. We identify dense cores at a number of selected epochs in the simulations at two density thresholds which roughly mimic the excitation densities of the NH3 (J - K) = (1,1) transition and the N2H+ (1-0) emission line. A noticeable global difference between the two simulations is the core formation efficiency (CFE) of the high-density cores. In the strongly supercritical simulations, the CFE is 33% per unit free-fall time of the cloud (t ff,cl), whereas in the mildly supercritical simulations this value goes down to ~6 per unit t ff,cl. A comparison of the intrinsic specific angular momentum (j 3D) distributions of the cores with the specific angular momentum derived using synthetic two-dimensional (2D) velocity maps of the cores (j 2D) shows that the synthetic observations tend to overestimate the true value of the specific angular momentum by a factor of ~8-10. We find that the distribution of the ratio j 3D/j 2D of the cores peaks at around ~0.1. The origin of this discrepancy lies in the fact that contrary to the intrinsic determination of j which sums up the individual gas parcels' contributions to the angular momentum, the determination of the specific angular momentum using the standard observational procedure which is based on a measurement on the global velocity gradient under the hypothesis of uniform rotation smoothes out the complex fluctuations present in the 3D velocity field. Our results may well provide a natural explanation for the discrepancy by a factor of ~10 observed between the intrinsic 3D distributions of the specific angular momentum and the corresponding distributions derived in real observations. We suggest that previous and future measurements of the specific angular momentum of dense cores which are based on the measurement of the observed global velocity gradients may need to be reduced by a factor of ~10 in order to derive a more accurate estimate of the true specific angular momentum in the cores. We also show that the exponent of the size-specific angular momentum relation is smaller (~1.4) in the synthetic observations than their values derived in the 3D space (~1.8).