In 1995, Komlós, Sárközy and Szemerédi showed that every large $n$-vertex graph with minimum degree at least $(1/2 + γ)n$ contains all spanning trees of bounded degree. We consider a generalization of this result to loose spanning hypertrees in 3-graphs, that is, linear hypergraphs obtained by successively appending edges sharing a single vertex with a previous edge. We show that for all $γ$ and $Δ$, and $n$ large, every $n$-vertex 3-uniform hypergraph of minimum vertex degree $(5/9 + γ)\binom{n}{2}$ contains every loose spanning tree $T$ with maximum vertex degree $Δ$. This bound is asymptotically tight, since some loose trees contain perfect matchings.