Progress in computing the spectrum of excited baryons and mesons in lattice QCD is described. Results in the zero-momentum bosonic I = 1/2, S = 1, T1u symmetry sector of QCD using a correlation matrix of 58 operators are presented. All needed Wick contractions are efficiently evaluated using a stochastic method of treating the low-lying modes of quark propagation that exploits Laplacian Heaviside quark-field smearing. Level identification using probe operators is discussed.