Summary: We establish some new sufficient conditions which guarantee that the solution \(x\) of a third-order neutral delay dynamic equation \[ \biggl(a(t)\biggl[\big(r(t)x(t)+p(t)x(g(t)))^\Delta\big)^\Delta\biggl]^\gamma\biggl)^\Delta +q(t)x\gamma(\tau(t))=0 \] is either oscillatory or satisfies \(\lim_{t\rightarrow\infty} x(t)=0\); here \(\gamma>0\) is a quotient of odd positive integers, \(a,r,p\), and \(q\) are real-valued positive rd-continuous functions defined on \(\mathbb T\) which is unbounded above. Our results, as a special case when \(\mathbb T=\mathbb R\), include some of the results in [B. Baculíková and J. Džurina, Math. Comput. Modelling 52, No. 1–2, 215–226 (2010; Zbl 1201.34097)], and improve some results obtained for third-order dynamic equations without a neutral term on time scales. In particular, the Hille-Nehari type criteria obtained not only extend some related results reported in [L. Erbe et al., J. Math. Anal. Appl. 329, No. 1, 112–131 (2007; Zbl 1128.39009)] but also improve those by S. H. Saker [Sci. China, Math. 54, No. 12, 2597–2614 (2011; Zbl 1260.34166)] and Y. Wang and Z. Xu [J. Comput. Appl. Math. 236, No. 9, 2354–2366 (2012; Zbl 1239.34112)] in the case of \(\gamma\geq 1\).