We study ideals generated by the (complements of) facets of d-dimensional "interval" simplicial complexes. This class recovers interval graphs when d=1, and strictly contains the class of pure shifted complexes. They also played a role in recent work of Benedetti, Seccia, and Varbaro in their study of determinantal facet ideals. We construct minimal cellular resolutions of such ideals, supported on a certain space of directed graph homomorphisms that generalize the "box of complexes resolutions" of Nagel and Reiner. We conclude that these ideals have linear resolutions, providing a generalization/specialization of Froberg's theorem regarding edge ideals of chordal graphs. Based on old joint work with Alex Engstrom, as well as more recent conversations with Bennet Goeckner and Marta Pavelka.