We develop a general framework for describing thermoelectric effects in phase-coherent superconducting structures. Formulas for the electrical conductance, thermal conductance, thermopower, and Peltier coefficient are obtained and their various symmetries discussed. Numerical results for both dirty and clean Andreev interferometers are presented. We predict that giant oscillations of the thermal conductance can occur, even when oscillations in the electrical conductance are negligibly small. Results for clean, two-dimensional systems with a single superconducting inclusion are also presented, which show that normal-state oscillations arising from quasiparticle boundary scattering are suppressed by the onset of superconductivity. In contrast, for a clean system with no normal-state boundary scattering, switching on superconductivity induces oscillations in off-diagonal thermoelectric coefficients.