We have analysed the long- and short-term time dependence of the pulse arrival times and the pulse detection rates for eight Rotating Radio Transient (RRAT) sources from the Parkes Multi-beam Pulsar Survey (PMPS). We find significant periodicities in the individual pulse arrival times from six RRATs. These periodicities range from 30 minutes to 2100 days and from one to 16 independent (i.e. non-harmonically related) periodicities are detected for each RRAT. In addition, we find that pulse emission is a random (i.e. Poisson) process on short (hour-long) time scales but that most of the objects exhibit longer term (months-years) non-random behaviour. We find that PSRs J1819-1458 and J1317-5759 emit more doublets (two consecutive pulses) and triplets (three consecutive pulses) than is expected in random pulse distributions. No evidence for such an excess is found for the other RRATs. There are several different models for RRAT emission depending on both extrinsic and intrinsic factors which are consistent with these properties. ; Comment: Accepted by MNRAS on 2011 July 2. Contains 11 pages, 4 figures, 4 tables