I construct MSBM as a generalization of skew Brownian motion to the case of infinitely many interfaces x_k, with k ranging over the integers. This process behaves like Brownian motion when away from the interfaces, and experiences a skewness (or localized drift) alpha_k at each x_k. The construction and most of the results are derived using the representation of MSBM as a scaling of Brownian motion under a random time change. Then the theory of Dirichlet forms is used to derive the L^2 semigroup of MSBM and connect it to a diffusion process with discontinuous coefficient. As an application, I give some results concerning advection-diffusion in a two dimensional layered medium, and an elementary proof of an arcsine law for skew Brownian motion.