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Results of Matveev, Piergallini, and Amendola show that any two triangulations of a three-manifold with the same number of vertices are related to each other by a sequence of local combinatorial moves (namely, 2-3 and 3-2 moves). For some applications however, we need our triangulations to have certain properties, for example that all edges are essential. (An edge is inessential if both ends are incident to a single vertex, into which the edge can be homotoped.) We show that if the universal cover of the manifold has infinitely many boundary components, then the set of essential ideal triangulations is connected under 2-3, 3-2, 0-2, and 2-0 moves. Our results have applications in veering triangulations and in quantum invariants such as the 1-loop invariant. This is joint work with Tejas Kalelkar and Saul Schleimer.