Nonuniform strain applied to graphene's honeycomb lattice can induce pseudo-Landau levels in the single-particle spectrum. Various generalizations have been put forward, including a particular family of hopping models in d space dimensions. Here we show that the key ingredient for sharp pseudo-Landau levels in higher dimensions is dimensional reduction. We consider particles moving on a d-dimensional hyperdiamond lattice which displays a semimetallic band structure, with a (d-2)-dimensional nodal manifold. By applying a suitable strain pattern, the single-particle spectrum evolves into a sequence of relativistic Landau levels. We develop and solve the corresponding field theory: Each nodal point effectively generates a Landau-level problem which is strictly two dimensional to leading order in the applied strain. While the effective pseudovector potential varies across the nodal manifold, the Landau-level spacing does not. Our theory paves the way for strain engineering of single-particle states via dimensional reduction and beyond global minimal coupling.