Increasing the efficiency and density of power electronic systems (PESs) is an important objective for many high-impact applications, such as electric vehicle charging and aircraft electrification. Due to compactness and high heat dissipation, careful thermal monitoring of such PESs is required. Strategic placement of temperature sensors can improve accuracy of real-time temperature distribution estimates. Enhanced temperature estimation supports increased power throughput and density because PESs can be operated in a less conservative manner while still preventing thermal failure. This article presents new methods for temperature sensor placement for 2- and 3-dimensional PESs that 1) improve computational efficiency (by orders of magnitude in at least one case), 2) support use of more accurate evaluation metrics, and 3) are scalable to high-dimension sensor placement problems. These methods are tested via sensor placement studies based on a single-phase Flying Capacitor Multi-Level (FCML) prototype inverter. Information-based metrics are derived from a Resistance-Capacitance (RC) lumped parameter thermal model. Other more general metrics and system models are possible through application of a new continuous relaxation strategy introduced here for placement representation. A new linear programming (LP) formulation is presented that is compatible with a particular type of information-based metric. This LP strategy is demonstrated to support efficient solution of finely-discretized large-scale placement problems. The optimal sensor locations obtained from these methods were tested via physical experiments. The new methods and results presented here may aid the development of thermally-aware PESs with significantly enhanced capabilities.