Several compelling experimental results for exotic hadrons have emerged over the past 15 years.  Some of these puzzling new states are interspersed among the standard charmonium mesons and others among the bottomonium mesons.  These exotic candidates are a difficult challenge for lattice QCD, so our group at York University has been investigating a particular type of doubly heavy tetraquark that is more amenable to lattice calculations.  In this talk, our recent results will be presented with an emphasis on the basic phenomenology rather than on lattice technicalities.