For a fixed finite set of finite tournaments F, the F-free orientation problem asks whether a given finite undirected graph G has an F-free orientation, i.e., whether the edges of G can be oriented so that the resulting digraph does not embed any of the tournaments from F. We prove that for every F this problem is in P or NP-complete. Our proof reduces the classification task to a complete complexity classification of the orientation completion problem for F, which is the variant of the problem above where the input is a directed graph instead of an undirected graph, introduced by Bang-Jensen, Huang, and Zhu [J. Graph Theory, 87 (2018), pp. 285-304]. Our proof uses results from the theory of constraint satisfaction and a result of Agarwal and Kompatscher [J. Symb. Log., 83 (2018), pp. 395-415] about infinite permutation groups and transformation monoids.