An implicit direct-time integration method for obtaining transient responses of general dynamic systems is described. The conventional Newmark method cannot be directly applied to state-space first-order differential equations, which contain no explicit acceleration terms. The method proposed here is the state-space Newmark method that incorporates the average velocity concept, and can be applied to an analysis of general dynamic systems that are expressed by state-space first-order differential equations. It is also readily coded into a program. Stability and accuracy analyses indicate that the method is numerically unconditionally stable like the conventional Newmark method, and has a period error of 2nd-order accuracy for small damping and 4th-order for large damping and an amplitude error of 2nd-order, regardless of damping. In addition, its utility and validity are confirmed by two application examples. The results suggest that the proposed state-space Newmark method based on average velocity be generally applied to the analysis of transient responses of general dynamic systems with a high degree of reliability with respect to stability and accuracy.