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Accurately capturing the observed mean period of ENSO in general circulation models (GCMs) is often challenging, and it is therefore useful to understand which parameters and processes affect this period. A computationally efficient simulation-based approach is used to extract both the dominant eigenvalues and corresponding eigenvectors of the linearized model from the Zebiak--Cane intermediate-complexity model of ENSO without having to directly construct the linearization. The sensitivity of the period to a variety of parameters is examined, including atmosphere--ocean coupling, atmospheric heating parameterization, thermocline depth zonal profile, western boundary reflection coefficient, atmospheric and ocean wave speeds or Rossby radii of deformation, ocean decay time, and the strength of the annual cycle. In addition to the sensitivity information, the spatial structures of the main fields (SST, thermocline thickness, and more) that are involved in period changes are obtained to aid in the physical interpretation of the sensitivities. There are three main time lags that together compose one-half of a model ENSO period: the Rossbyplus-Kelvin wave propagation time for a wind-caused central Pacific disturbance to propagate to the western ocean and back, SST dynamics that determine the lag between eastern ocean thermocline anomalies and eastern ocean SST anomalies, and the 'accumulation' lag of integrating a sufficient delayed wave signal arriving from the western ocean to cancel the eastern ocean anomalies. For any of the parameter changes considered, the eigenvector changes show that the largest contributor to the period change is from changes to the last of these three mechanisms. Physical mechanisms that affect this accumulation delay are discussed, and the case is made that any significant change to ENSO's period is in turn likely to involve changes to this delay.