We show that the low-energy density of quasiparticle states in the mixed state of ultra-clean d-wave superconductors is characterized by pronounced quantum oscillations in the regime where the cyclotron frequency $hbaromega_c ll Delta_0$, the d-wave pairing gap. Such oscillations as a function of magnetic field B are argued to be due to the internodal scattering of the d-wave quasiparticles near wavevectors $(pm k_D,pm k_D)$ by the vortex lattice as well as their Zeeman coupling. The periodicity of the oscillations is set by the condition $k_D sqrt{hc/(eB)} equiv k_D sqrt{hc/(eB)}pmod {2pi}$. We find that there is additional structure within each period which grows in complexity as the Dirac node anisotropy increases.
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