We introduce a scheme that exploits laser cooling and phonon-mediated spin-spin interactions in crystals of trapped atomic ions to explore the transport of energy through a quantum magnet. We show how to implement an effective transport window to control the flow of energy through the magnet even in the absence of fermionic statistics for the carriers. This is achieved by shaping the density of states of the effective thermal reservoirs that arise from the interaction with the external bath of the modes of the electromagnetic field, and can be experimentally controlled by tuning the laser frequencies and intensities appropriately. The interplay of this transport window with the spin-spin interactions is exploited to build an analogue of the Coulomb-blockade effect in nano-scale electronic devices, and opens new possibilities to study quantum effects in energy transport.