In the present paper, we first develop the operational calculus of Mikusi\'{n}ski's type for the Caputo fractional differential operator. This calculus is used to obtain exact solutions of an initial value problem for linear fractional differential equations with constant coefficients and fractional derivatives in Caputo's sense. The initial conditions are given in terms of the field variable and its derivatives of integer order. The obtained solutions are expressed through Mittag-Leffler type functions. Special cases and integral representations of solutions are presented.