This paper attempts to study the risk-sensitive discounted continuous-time Markov decision processes with unbounded transition and cost rates. Different from the case of bounded transition/cost rates, the optimality equation (OE) no longer has a solution satisfying the uniform convergence condition introduced in the existing literature. Thus, we first replace the uniform convergence condition of the solution with a new boundary condition. Then, we find mild conditions imposed on the primitive data of the decision processes, which not only ensure the existence of a solution to the OE but also are the generalization of the bounded transition/cost rates conditions. Furthermore, using the characterization of the boundary condition and a novel technique, from the OE we prove the existence of an optimal policy out of the class of randomized history-dependent policies. Finally, we present two examples with unbounded transition/cost rates to illustrate our results.