The paper deals with piecewise linear differential systems on \(\mathbb R^n\) of the forms \(\dot x=A^{-}x+c\) for \(x_1\leq 0\) and \(\dot x=A^{+}x+c\) for \(x_1>0\), where \(x=(x_1,x_2,x_3)\in \mathbb R^{3}\), \(A^{\pm}\) are real \(3\times 3\) matrices, and \(c\in \mathbb R^{3}\). Assuming an observability of those systems, they are transformed to simple linear normal forms. Then those normalized systems are studied by showing existence or nonexistence results on periodic solutions along with their stability properties. To this end, Poincaré mappings of periodic solutions are introduced.