We investigate the quasiparticle density of states in disordered d-wave superconductors. By constructing a quantum map describing the quasiparticle dynamics in such a medium, we explore deviations of the density of states from its universal form (∝E), and show that additional low-energy quasiparticle states exist provided (i) the range of the impurity potential is much larger than the Fermi wavelength (allowing one to use recently developed semiclassical methods), (ii) classical trajectories exist along which the pair potential changes sign, and (iii) the diffractive scattering length is longer than the superconducting coherence length. In the classically chaotic regime, universal random matrix theory behavior is restored by quantum dynamical diffraction which shifts the low-energy states away from zero energy, and the quasiparticle density of states exhibits a linear pseudogap below an energy threshold E*≪Δ₀, much smaller than the superconducting gap.