Three-dimensional symmetric piecewise linear differential systems near the conditions corresponding to the fold-Hopf bifurcation for smooth systems are considered. By introducing one small parameter, we study the bifurcation of limit cycles in passing through its critical value, when the three eigenvalues of the linear part at the origin are at the imaginary axis of the complex plane. The simultaneous bifurcation of three limit cycles is proved. Conditions for stability of these limit cycles are provided, and analytical expressions for their period and amplitude are obtained. Finally, we apply the achieved theoretical results to a generalized version of Chua’s circuit, showing that the fold-Hopf bifurcation takes place for a certain range of parameters.