The first numerical predictions of the dynamical diquark model of multiquark exotic hadrons are presented. Using Born-Oppenheimer potentials calculated from lattice QCD and phenomenological diquark(triquark) masses, mass eigenvalues that are degenerate in spin and isospin are computed from numerical solutions to both coupled and uncoupled Schroedinger equations. Assuming reasonable estimates of the fine-structure splittings, we find that the band structure of our mass spectra agrees well with the experimentally observed spectrum of charmonium-like states. Using our best fits, we predict a number of unobserved states, such as pentaquark states that lie below the charmonium-plus-nucleon threshold.