Semiclassical approximations for quantum time correlation functions are presented for both electronically adiabatic and nonadiabatic dynamics along with discussions of the operator ordering and the classical limit. With the combined use of the initial-value representation of the semiclassical propagator, a discrete algorithm to evaluate the Jacobi matrices, semiclassical operator ordering rules, and the stationary-phase filter technique, a practical algorithm is developed to calculate quantum time correlation functions. This approach holds considerable promise for simulating the quantum dynamics of realistic many-body systems. Some simple illustrative examples are used to demonstrate the feasibility and accuracy of the algorithm.