In 1992 Agronsky and Ceder proved that any finite collection of non-degenerate Peano continua in the unit square is an ω-limit set for a continuous map. We improve this result by showing that it is valid, with natural restrictions, for the triangular maps (x,y) ↦ (f(x),g(x,y)) of the square. For example, we show that a non-trivial Peano continuum C ⊂ I² is an orbit-enclosing ω-limit set of a triangular map if and only if it has a projection property. If C is a finite union of Peano continua then, in addition, a coherence property is needed. We also provide examples of two slightly different non-Peano continua C and D in the square such that C is and D is not an ω-limit set of a triangular map. In view of these examples a characterization of the continua which are ω-limit sets for triangular mappings seems to be difficult.