We report the implementation of connected triple excitations in the equation-of-motion (EOM) coupled-cluster (CC) method for excitation energies for the first time. The reference state is described by the complete CC singles, doubles, and triples (CCSDT) method. Excited states are generated from the reference state wave function by the action of a linear excitation operator including single, double, and triple excitations. The excited state wave functions and energies are obtained by diagonalizing the effective Hamiltonian e(-T)He(T), where T is the cluster operator for the reference state, in the space of singly, doubly, and triply excited determinants. Comparison is made with full configuration interaction excitation energies for several examples (CH+, Be, SiH2, and CH2). These show that EOM-CCSDT is able to describe states which are doubly excited relative to the reference state, as well as singly excited states. Calculations of several excitation energies of BH using an extended basis set are also reported, and show good agreement with experiment.