We present in this paper a linear response theory based on our recently developed state-specific multireference coupled cluster (SS-MRCC) method to compute excited state energies for systems whose ground state has a pronounced multireference character. The SS-MRCC method is built on complete active space reference functions, and is designed to treat quasidegeneracy of varying degrees while bypassing the intruder problem. The linear response theory based on such a function [multireference coupled cluster based linear response theory (MR-CCLRT)] offers a very convenient access to computation of excited states and, in particular, to generation of potential energy surfaces (PES) for excited states where a traditional response formulation based on a single reference theory will fail due to the quasidegeneracy at some regions of the PES and the effective Hamiltonian-based multireference response methods would be plagued by intruders. An attractive feature of the MR-CCLRT is that the computed excitation energies are size intensive in the sense that they become asymptotically equal to the sum of fragment excitation energies in the limit of noninteracting fragments. Illustrative numerical results are presented for the excited state PES of the rectangular H-4 (P4) model, the trapezoidal H-4 (H4) model, for Li-2, and for some sample points on the excited states PES of the BeH2 complex. The ground states of all the three examples possess quasidegeneracy at some point on the PES, and there are potential intruders at some other points in the PES, and hence are appropriate to test the efficacy of the MR-CCLRT. A comparison with the (CI) full configuration interaction and MR-CCLRT results in the same basis for all the molecules shows very good performance of the theory in general, and indicates the efficacy of the method.