Let n be an even integer. We study the bifurcation of limit cycles from the periodic orbits of the n-dimensional linear center given by the differential system<(x)over dot>(1) = -x(2), <(x)over dot>(2) = x(1), ... , <(x)over dot>(n-1) = -x(n), <(x)over dot>(n) = x(n-1),perturbed inside a class of piecewise linear differential systems. Our main result shows that at most (4n - 6)(n/2-1) limit cycles can bifurcate up to first-order expansion of the displacement function with respect to a small parameter. For proving this result we use the averaging theory in a form where the differentiability of the system is not needed. (C) 2011 Elsevier Ltd. All rights reserved.Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)