A new explicit trigonometrically-fitted modified Runge-Kutta method (TFMRK) is developed for the numerical integration of first-order initial value problems (IVPs) with oscillatory solutions. The newly developed method was made according to the method of Runge-Kutta Dormand and to the third algebraic order. Linear stability of the new method are examined. Numerical results are reported to show the robustness and competence of the new method compared with other existing Runge-Kutta (RK) methods.