The aim of this paper is to determine the algebraic structure of multidimensional cyclic codes over a finite chain ring $\mathfrak{R}$. An algorithm to find the generator polynomials of $n$ dimensional ($n$D) cyclic codes of length $m_{1}m_{2}\dots m_{n}$ over $\mathfrak{R}$ has been developed using the generator polynomials of cyclic codes over $\mathfrak{R}$. Additionally, the generators of $n$D cyclic codes with length $m_{1}m_{2}\dots m_{n}$ over $\mathfrak{R}$ have been obtained as separable polynomials for the case $q\equiv 1(mod~ m_{j}), j\geq 2$, where $q=p^{r}$ is the cardinality of residue field of $\mathfrak{R}$.