We present a variant of a universality result of Roth (1986) for sparse, 3-uniform hypergraphs contained in strongly jumbled hypergraphs. One of the ingredients of our proof is a counting lemma for fixed hypergraphs in sparse ``pseudorandom{''} hypergraphs, which is proved in the companion paper (Counting results for sparse pseudorandom hypergraphs I). (C) 2017 Elsevier Ltd. All rights reserved.