In this paper, we develop a new class of linear time-integration schemes for phase-field models. The newly proposed schemes extend the recently developed energy quadratization technique by introducing extra free parameters to further stabilize the schemes and improve their accuracy. The freshly proposed schemes have several advantages. First of all, they are rather generic such that they apply to most existing phase-field models in the literature. The resulted schemes are also linear in time, which means only a linear system needs to be solved during each time marching step. Thus, it significantly reduces the computational cost. Besides, they are unconditionally energy stable such that a larger time step size is practical. What is more, the solution existence and uniqueness in each time step are guaranteed without any dependence on the time step size. To demonstrate the generality of the proposed schemes, we apply them to several typical examples, including the widely-used molecular beam epitaxy (MBE) model, the Cahn-Hilliard equation, and the diblock copolymer model. Numerical tests reveal that the proposed schemes are accurate and efficient. This new family of linear and unconditionally energy stable schemes provides insights in developing numerical approximations for general phase field models.