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In this paper, a new computational method based on the second kind Chebyshev wavelets (SKCWs) together with the Galerkin method is proposed for solving a class of stochastic heat equation. For this purpose, a new stochastic operational matrix for the SKCWs is derived. A collocation method based on block pulse functions is employed to derive a general procedure for forming this matrix. The SKCWs and their operational matrices of integration and stochastic Itô-integration are used to transform the under consideration problem into the corresponding linear system of algebraic equations which can be simply solved to achieve the solution of the problem. The proposed method is very convenient for solving such problems, since the initial and boundary conditions are taken into account automatically. Moreover, the efficiency of the proposed method is shown for some concrete examples. The results reveal that the proposed method is very accurate and efficient. [ABSTRACT FROM AUTHOR]