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A conjecture by Corvaja and Zannier predicts that smooth, projective, simplyconnected varieties over a number field with Zariski dense set of rationalpoints have the Hilbert Property; this was proved by Demeio for Kummer surfaceswhich are associated to products of two elliptic curves.  In this article, over a finitely generated field of characteristic zero, weestablish the Hilbert Property for all Kummer surfaces associated to theJacobian of a genus $2$ curve. In general we prove that all Jacobian Kummervarieties associated to a hyperelliptic curve of genus $\geq 2$ of odd degreealso have the Hilbert Property.

Rational curves and the Hilbert Property on Jacobian Kummer varieties