We present a linear-response formulation of density cumulant theory (DCT) that provides a balanced and accurate description of many electronic states simultaneously. In the original DCT formulation, only information about a single electronic state (usually, the ground state) is obtained. We discuss the derivation of linear-response DCT, present its implementation for the ODC-12 method (LR-ODC-12), and benchmark its performance for excitation energies in small molecules (N$_2$, CO, HCN, HNC, C$_2$H$_2$, and H$_2$CO), as well as challenging excited states in ethylene, butadiene, and hexatriene. For small molecules, LR-ODC-12 shows smaller mean absolute errors in excitation energies than equation-of-motion coupled cluster theory with single and double excitations (EOM-CCSD), relative to the reference data from EOM-CCSDT. In a study of butadiene and hexatriene, LR-ODC-12 correctly describes the relative energies of the singly-excited $1^1mathrm{B_{u}}$ and the doubly-excited $2^1mathrm{A_{g}}$ states, in excellent agreement with highly accurate semistochastic heat-bath configuration interaction results, while EOM-CCSD overestimates the energy of the $2^1mathrm{A_{g}}$ state by almost 1 eV. Our results demonstrate that linear-response DCT is a promising theoretical approach for excited states of molecules.
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