Thermally induced spin-dependent transport across magnetic tunnel junctions (MTJs) is theoretically investigated. We analyze the thermal analog of Slonczewski's model (as well as its limiting case-Julliere's model) of tunneling magnetoresistance and obtain analytical expressions for the junction thermopower and the tunneling magnetothermopower (TMT). The analytical model is tested numerically for the special case of an Al2O3-based MTJ, for which we analyze the dependence of the thermopower and TMT on the relative magnetization orientations, as well as on the barrier height and thickness. We show that at a certain barrier height TMT vanishes, separating the region of positive and negative TMT. As its electrical prototype, this thermal spin transport model should serve as a phenomenological benchmark for analyzing experimental and first-principles calculations of thermopower in magnetic tunnel junctions. The analytical expressions can be used as a first estimate of the magnetothermopower of the junctions using ab initio band structure data of the junction ferromagnets.