A Monte Carlo method, namely, the "statistical counting method" (SCM) has been proposed for simulating the conformational entropy of a single free or confined linear self-avoiding random walk chains on the simple cubic lattice. For a free linear chain with length 1081, it is found from the calculated results of 100 groups of 10(4) samplings that the maximum and the minimum values of the conformational entropy are 1670.2 and 1660.3, respectively, the deviations from the average value 1663.8 are only +0.39% and -0.2%. In the range of chain length 8-20 the calculated entropy data are found to be in agreement with their precise values obtained by M. F. Sykes, et al. [J. Phys. A 5, 653 (1972)] with deviation less than 0.05%. In the range of chain lengths up to 27 confined in a cube of side length 2, the entropy data are also consistent with their precise values obtained from the direct counting of conformation number with the deviations less than 0.6%. For a long chain with lengths up to 2101, the calculated entropy data have confirmed the prediction by the renormalization group theory very well and the deviation is less than 0.8%.