Numerical techniques have predicted that reentrant electrical scroll waves underlie many cardiac arrhythmias, but experimental limitations have hampered a detailed understanding of the specific mechanisms responsible for reentrant wave formation and breakup. To further this effort, we recently have begun to apply the technique of data assimilation, widely used in weather forecasting, to reconstruct time series in cardiac tissue. Here we use model-generated synthetic observations from a numerical experiment to evaluate the performance of the ensemble Kalman filter in reconstructing such time series for a discordant alternans state in one spatial dimension and for scroll waves in three dimensions. We show that our approach is able to recover time series of both observed and unobserved variables that match the truth. Where nearby observations are available, the error is reduced below the synthetic observation error, with a smaller reduction with increased distance from observations. Using one-dimensional cases, we provide a deeper analysis showing that limitations in model formulation, including incorrect parameter values and undescribed spatial heterogeneity, can be managed appropriately and that some parameter values can be estimated directly as part of the data assimilation process. Our findings demonstrate that state reconstruction for spatiotemporally complex cardiac electrical dynamics is possible and has the potential for successful application to real experimental data.