A recursive filtering technique for the state estimation of linear systems where the Gaussian assumption is not required for either the plant (process), initial condition, or measurement noise are presented. The approach requires the noise to be defined by their higher-order statistics (moments or cumulants). Analogous to the Kalman filter time-update propagation of a covariance matrix, cumulants can be propagated in a similar fashion making use of Kronecker products and the fact that the cumulant of a sum of independent random variables is the sum of the individual cumulants. This allows for a straightforward characterization of the statistics for the likelihood and the prior probability at the time of the measurement update of the filter. An experimental evaluation shows these cumulant-based filters perform better than a default use of a Kalman filter.