The problem of mean thermal radiation transfer in a fluctuating, statistically uniform medium, with spatial temperature distribution given by a generalized telegraphic process, is solved. The solution is based on the use of the Laplace transformation and replacement of an original temperature-fluctuation distribution function by another one, which practically does not distort average terms in the equations for radiation. For this function, we use a superposition of Dirac delta-functions which allow the inverse Laplace transformation to be made analytically. Increasing the number of terms in the superposition, we reduce the error of the solution. The solution obtained is used to estimate the accuracy of the optically-thin pulsation approximation in the theory of radiation transfer in turbulent flows. A case of large fluctuations of the absorption coefficient and Planck function is analyzed. It is shown that the outgoing mean radiation as a function of layer optical thickness has a maximum arising due to the fluctuations. This anomalous dependence may cause a dip in the center of the spectral line radiation profile, even if the layer is statistically uniform.