In the lattice structure considered here, crack propagation is caused by feeding waves, carrying energy to the crack front, and accompanied by dissipative waves carrying a part of this energy away from the front (the difference is spent on the bond disintegration). The feeding waves differ by their wavenumber. A zero feeding wavenumber corresponds to a macrolevel-associated solution with the classical homogeneous-material solution as its long-wave approximation. A non-zero wavenumber corresponds to a genuine microlevel solution which has no analogue on the macrolevel. In the latter case, on the crack surfaces and their continuation, the feeding wave is located behind (ahead) the crack front if its group velocity is greater (less) than the phase velocity. Dissipative waves, which appear in both macrolevel-associated and microlevel solutions, are located in accordance with the opposite rule. (Wave dispersion is the underlying phenomenon which allows such a wave configuration to exist.) In contrast to a homogeneous material model, both these solutions permit supersonic crack propagation. Such feeding and dissipative waves and other lattice phenomena are characteristic of dynamic phase transformation as well. In the present paper, mode III crack propagation in a square-cell elastic lattice is studied. Along with the lattice model, some simplified one-dimensional structures are considered allowing one to retrace qualitatively (with no technical difficulties) the main lattice phenomena.