In this paper, we present a class of three-dimensional dynamical systems having multiscrolls which we call unstable dissipative systems (UDSs). The UDSs are dissipative in one of its components but unstable in the other two. This class of systems is constructed with a switching law to display various multiscroll strange attractors. The multiscroll strange attractors result from the combination of several unstable "one-spiral" trajectories by means of switching. Each of these trajectories lies around a saddle hyperbolic stationary point. Thus, we describe how a piecewise-linear switching system yields multiscroll attractors, symmetric or asymmetric, with chaotic behavior.