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In the previous study, one of the authors investigated stationary response characteristics of linear systems subjected to non-Gaussian random excitation. It was revealed that the probability distribution of the stationary response varies significantly depending upon the bandwidth of the excitation power spectrum. In real engineering problems, on the other hand, the transient response before reaching the stationary state is also crucial. Therefore, in this paper, we explore transient response characteristics to non-Gaussian random excitation by Monte Carlo simulation. The non-Gaussian excitation is prescribed by both a probability density function and a power spectrum with bandwidth parameter. In order to find the response characteristics that appears commonly for various non-Gaussian excitation, we consider four kinds of non-Gaussian distributions for the probability distributions of the excitations. Their shapes are much different from a Gaussian distribution and also distinct from each other. For such non-Gaussian random excitations, we look into the relationship between the transient response characteristics and the excitation bandwidth. Specifically, first, the transient response distribution is examined. Then, we observe the variance, skewness and kurtosis of the transient response to discuss the time variation of the response distribution in more detail.