The rotational properties of centrally condensed, turbulent molecular cloud cores with velocity fields that are characterized by Gaussian random fields are investigated. It is shown that the observed line width-size relationship can be reproduced if the velocity power spectrum is a power-law with P(k)=k**n and n=-3 to -4. The line-of-sight velocity maps of these cores show velocity gradients that can be interpreted as rotation. For n=-4, both, the deduced values of the angular velocity

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... km/s/pc * (R/0.1 pc)**0.5 and the scaling relations between Omega and the core radius R are in very good agreement with the observations. As a result of the dominance of long wavelength modes, the cores also have a net specific angular momentum with an average value of j=7*(10**20)*(R/0.1 pc)**(1.5) cm**2/s with a large spread. Their internal dimensionless rotational parameter is beta=0.03, independent of the scale radius R. In general, the line-of-sight velocity gradient of an individual turbulent core does not provide a good estimate of its internal specific angular momentum. We find however that the distribution of the specific angular momenta of a large sample of cores which are described by the same power spectrum can be determined very accurately from the distribution of their line-of-sight velocity gradients Omega using the simple formula j=p*Omega*R*R where p depends on the density distribution of the core and has to be determined from a Monte-Carlo study. Our results show that for centrally condensed cores the intrinsic angular momentum is overestimated by a factor of 2-3 if p=0.4 is used.