  In this work, we study spacelike surfaces in Minkowski space $E_1^3$ foliated
by pieces of circles and that satisfy a linear Weingarten condition of type $a
H+b K=c$, where $a,b$ and $c$ are constant and $H$ and $K$ denote the mean
curvature and the Gauss curvature respectively. We show that such surfaces must
be surfaces of revolution or surfaces with constant mean curvature H=0 or
surfaces with constant Gauss curvature K=0.
