A large number of the presently running neutrino and dark-matter experiments use thallium-doped cesium-iodide CsI[Tl] crystals, sodium-doped cesium-iodide CsI[Na] crystals, or thallium-doped sodium-iodide NaI[Tl] crystals. In the present paper we calculate elastic and inelastic cross sections for neutral-current supernova-neutrino scattering off $^{127}\mathrm{I}$ and $^{133}\mathrm{Cs}$, relevant for experiments using CsI[Tl], CsI[Na], or NaI[Tl] crystals. We study also the cross sections folded with two-parameter Fermi-Dirac distributions of the supernova-neutrino spectrum. The adopted nuclear-theory framework is the microscopic quasiparticle-phonon model, able to operate in large single-particle valence bases and use G-matrix based two-nucleon interactions with a phenomenological renormalization.