  We present a detailed study of elliptic fibrations on Fourier-Mukai partners
of K3 surfaces, which we call derived elliptic structures. We fully classify
derived elliptic structures in terms of Hodge-theoretic data, similar to the
Derived Torelli Theorem that describes Fourier-Mukai partners. In Picard rank
two, derived elliptic structures are fully determined by the Lagrangian
subgroups of the discriminant group. As a consequence, we prove that for a
large class of Picard rank 2 elliptic K3 surfaces all Fourier-Mukai partners
are Jacobians, and we partially extend this result to non-closed fields. We
also show that there exist elliptic K3 surfaces with Fourier-Mukai partners
which are not Jacobians of the original K3 surface. This gives a negative
answer to a question raised by Hassett and Tschinkel.

Algebraic Geometry;K-Theory and Homology