In this paper we study the structure of $θ$-cyclic codes over the ring $B_k$ including its connection to quasi-$\tildeθ$-cyclic codes over finite field $\mathbb{F}_{p^r}$ and skew polynomial rings over $B_k.$ We also characterize Euclidean self-dual $θ$-cyclic codes over the rings. Finally, we give the generator polynomial for such codes and some examples of optimal Euclidean $θ$-cyclic codes.