Phase retrieval refers to the problem of recovering real- or complex-valued vectors from magnitude measurements. The best-known algorithms for this problem are iterative in nature and rely on so-called spectral initializers that provide accurate initialization vectors. We propose a novel class of estimators suitable for general nonlinear measurement systems, called linear spectral estimators (LSPEs), which can be used to compute accurate initialization vectors for phase retrieval problems. The proposed LSPEs not only provide accurate initialization vectors for noisy phase retrieval systems with structured or random measurement matrices, but also enable the derivation of sharp and nonasymptotic mean-squared error bounds. We demonstrate the efficacy of LSPEs on synthetic and real-world phase retrieval problems, and we show that our estimators significantly outperform existing methods for structured measurement systems that arise in practice.