Random walks on left regular bands were first introduced and analyzed by \textit{K. S. Brown} [J. Theor. Probab. 13, No. 3, 871--938 (2000; Zbl 0980.60014)]. A special type of left regular bands are the semigroup of faces of a hyperplane arrangement. In this article, the author presents a simple construction of eigenvectors for the transition matrices of random walks on left regular bands. This is achieved by a decomposition of the transitional matrices into a linear combinations of orthogonal idempotents by specializing a simple recursive method introduced by the author [Int. J. Algebra Comput. 17, No. 8, 1593--1610 (2007; Zbl 1148.16024)].This method provides a simpler alternative prove to some known results on random walks on left regular bands. The author also explain at the end of the paper how to use the poset topology to extract an eigenbasis for the transition matrices of the hyperplane chamber walks.