The microwave radiation used to heat food is one type of non-ionizing radiation. In general any electromagnetic radiation with a photon energy less than about 10 electron volts is considered to be non-ionizing radiation. This boundary falls in the ultraviolet and very roughly corresponds to the amount of energy it typically takes to ionize an atom. For example, visible light, microwaves, and radio waves are non-ionizing, while hard ultraviolet, x-rays, and gamma rays are ionizing.
More generally, the answer depends on how you define "particle" and "radiation". For example, acoustic radiation such as sound or ultrasound is non-ionizing, but "phonons" are usually considered to be "only" quasiparticles. (The distinction between particle and quasiparticle, however, depends on one's theoretical framework.) Cosmic background neutrinos and gravitational radiation are two types of non-ionizing radiation that permeate all of space (including human bodies). Neutrinos are definitely particles, but without an established quantum theory of gravity, gravitons are only hypothetical.
Neutron radiation does not directly ionize atoms, but is deadly because it induces ionizing radiation in the form of nuclei recoiling from neutron scattering, gamma rays from neutron excitation of nuclei, or alpha, beta, or gamma radiation from neutron activation.