Do current general circulation models model climate oscillations?
How the models do at this sort of interannual variability for the current climate is described in Section 9.5 of the Fifth Assessment Report (AR5). For example, the section about ENSO says,


  The representation of ENSO in climate models has steadily improved and
  now bears considerable similarity to observed ENSO properties.
  However, as was the case in the AR4, simulations of both background climate (time mean and seasonal cycle, see Section 9.4.2.5.1) and internal variability exhibit serious systematic errors, many of which can be traced to the representation of deep convection, trade wind strength and cloud feedbacks, with little improvement from CMIP3 to CMIP5.
  
  While a number of CMIP3 models do not exhibit an ENSO variability maximum at the observed 2- to 7- year time scale, most CMIP5 models do have a maximum near the observed range and fewer models have the tendency for biennial oscillations. In CMIP3 the amplitude of El Niño ranged from less than half to more than double the observed amplitude. By contrast, the CMIP5 models show less inter-model spread (Figure 9.36). The CMIP5 models still exhibit errors in ENSO amplitude, period, irregularity, skewness, spatial patterns or teleconnections.
  
  ...
  
  Detailed quantitative evaluation of ENSO performance is hampered by the short observational record of key processes and the complexity and diversity of the processes involved. While shortcomings remain, the CMIP5 model ensemble shows some improvement compared to CMIP3, but there has been no major breakthrough and the multi-model improvement is mostly due to a reduced number of poor-performing models.


In short, it's a bit of a mixed bag and it will depend on which metrics you use to assess the variability.  Something like the NINO3.4 index is simple to calculate, but getting a good (or bad) score doesn't necessarily indicate how well your model simulates the relevant processes (e.g., ocean mixed layer dynamics, air-sea interactions, convection).

Section 9.5 also has information on other modes of variability, including the Pacific Decadal Oscillation, the Madden-Julian Oscillation, Atlantic Meridional Overturning Circulation, etc.  It's worth a read.