We report molecular dynamics (MD) simulation of energy exchange between single-walled carbon nanotubes (CNTs) and two aprotic solvents, acetonitrile and cyclohexane. Following our earlier study of hydrated CNTs, we find that the time scales and molecular mechanisms of the energy transfer are largely independent of the nature of the surrounding medium, and therefore, should hold for other media including polymer matrices and DNA. The vibrational energy exchange between CNT and solvents exhibits two time-scales. Over half of the energy is transferred in less than one picosecond, indicating that the dominant exchange mechanism is inertial relaxation. It occurs by collisions of solvent molecules with CNT walls, facilitated by the short-range Lennard-Jones interaction. Additional several picoseconds are required for the remainder of the vibrational energy exchange, corresponding to the diffusive relaxation mechanism and involving collective molecular motions. The faster stage of the CNT-solvent energy exchange occurs on the same time-scale, and therefore, competes with the vibrational energy relaxation inside CNTs. The energy exchange time-scales are significantly influenced by the arrangement of solvent molecules inside CNTs. Generally, the effects of confinement on the dynamics can be rationalized by analysis of the solvent structure. For the same CNT diameter, the extent of the confinement effect strongly depends on the size of the solvent molecules. Icelike properties in water seen in small CNTs disappear in CNTs with intermediate diameters. In acetonitrile and cyclohexane, medium size CNTs still show strong confinement effects. Rotational motions of acetonitrile molecules are inhibited, and the cyclohexane density is dramatically decreased. The disbalance between the local temperatures of the inside and outside regions of the solvent equilibrates through a tube-mediated interaction, rather than by a direct coupling between the two solvent subsystems. In all cases, the CNT-solvent energy transfer is mediated by slow motions in the frequency range of CNT radial breathing modes.