Third-order nonlinear difference equations of the form Δ(cnΔ(dnΔxn)) + pnΔxn+1 + qnf (xn−σ )= 0 ,n ≥ n0 are considered. Here, {cn} , {dn} , {pn} ,a nd{qn} are sequences of positive real numbers for n0 ∈ N, f is a continuous function such that f (u) /u ≥ K> 0f oru � 0 . By means of a Riccati transformation technique we obtain some new oscillation criteria. Examples are given to illustrate the importance of the results.