Non-Gaussian normal diffusion has been observed in many systems.Contradicting the common belief that the normal diffusion is associated with a Gaussian distribution,the spat ial displacements are distributed with an exponential tail as increasing the observation time.The current interpretation of this phenomena mostly is based on the heuristic notion of diffusing diffusivity,where the environmental fluctuations are modeled by means of a random diffusion constant D(t).However,the microscopic mechanism is not clear.Recently,we study the diffusion of an overdamped Brownian particle in a quasi-1D corrugated channel with fluctuating width.Without having recourse to paradigm of diffusing diffusion,we recover the essentials of the non-Gaussian normal diffusion: The distribution of the particle displacements along the channel is Gaussian for observation times either very short(local diffusion),or larger than the correlation time of the channel fluctuation,and exponential for a rather wide interval of intermediate observation times; For intermediate to large observation times the particle undergoes normal diffusion with one and the same effective diffusion constant.This simple microscopic model gives a clue to underlying the mechanisms.