Pseudorandom functions (PRFs) are one of the fundamental building blocks in cryptography. Traditionally, there have been two main approaches for PRF design: the "practitioner's approach" of building concretely-efficient constructions based on known heuristics and prior experience, and the "theoretician's approach" of proposing constructions and reducing their security to a previously-studied hardness assumption. While both approaches have their merits, the resulting PRF candidates vary greatly in terms of concrete efficiency and design complexity. In this work, we depart from these traditional approaches by exploring a new space of plausible PRF candidates. Our guiding principle is to maximize simplicity while optimizing complexity measures that are relevant to cryptographic applications. Our primary focus is on weak PRFs computable by very simple circuits-depth-2 ACC 0 circuits. Concretely, our main weak PRF candidate is a "piecewise-linear" function that first applies a secret mod-2 linear mapping to the input, and then a public mod-3 linear mapping to the result. We also put forward a similar depth-3 strong PRF candidate. The advantage of our approach is twofold. On the theoretical side, the simplicity of our candidates enables us to draw many natural connections between their hardness and questions in complexity theory or learning theory (e.g., learnability of depth-2 ACC 0 circuits and width-3 branching programs, interpolation and property testing for sparse polynomials, and natural proof barriers for showing super-linear circuit lower bounds). On the applied side, the piecewise-linear structure of our candidates lends itself nicely to applications in secure multi-party computation (MPC). Using our PRF candidates, we construct protocols for distributed PRF evaluation that achieve better round complexity and/or communication complexity (often both) compared to protocols obtained by combining standard MPC protocols with PRFs like AES, LowMC, or Rasta (the latter two are specialized MPC-friendly PRFs). Finally, we introduce a new primitive we call an encoded-input PRF, which can be viewed as an interpolation between weak PRFs and standard (strong) PRFs. As we demonstrate, an encoded-input PRF can often be used as a drop-in replacement for a strong PRF, combining the efficiency benefits of weak PRFs and the security benefits of strong PRFs. We conclude by showing that our main weak PRF candidate can plausibly be boosted to an encoded-input PRF by leveraging standard error-correcting codes.