Differential equations with distributional sources---in particular, involving delta distributions and/or derivatives thereof---have become increasingly ubiquitous in numerous areas of physics and applied mathematics. It is often of considerable interest to obtain numerical solutions for such equations, but the singular ("point-like") modeling of the sources in these problems typically introduces nontrivial obstacles for devising a satisfactory numerical implementation. A common method to circumvent these is through some form of delta function approximation procedure on the computational grid, yet this strategy often carries significant limitations. In this paper, we present an alternative technique for tackling such equations: the "Particle-without-Particle" method. Previously introduced in the context of the self-force problem in gravitational physics, the idea is to discretize the computational domain into two (or more) disjoint pseudospectral (Chebyshev-Lobatto) grids in such a way that the "particle" (the singular source location) is always at the interface between them; in this way, one only needs to solve homogeneous equations in each domain, with the source effectively replaced by jump (boundary) conditions thereon. We prove here that this method is applicable to any linear PDE (of arbitrary order) the source of which is a linear combination of one-dimensional delta distributions and derivatives thereof supported at an arbitrary number of particles. We furthermore apply this method to obtain numerical solutions for various types of distributionally-sourced PDEs: we consider first-order hyperbolic equations with applications to neuroscience models (describing neural populations), parabolic equations with applications to financial models (describing price formation), second-order hyperbolic equations with applications to wave acoustics, and finally elliptic (Poisson) equations.