  Relative Heffter arrays, denoted by $\mathrm{H}_t(m,n; s,k)$, have been
introduced as a generalization of the classical concept of Heffter array. A
$\mathrm{H}_t(m,n; s,k)$ is an $m\times n$ partially filled array with elements
in $\mathbb{Z}_v$, where $v=2nk+t$, whose rows contain $s$ filled cells and
whose columns contain $k$ filled cells, such that the elements in every row and
column sum to zero and, for every $x\in \mathbb{Z}_v$ not belonging to the
subgroup of order $t$, either $x$ or $-x$ appears in the array. In this paper
we show how relative Heffter arrays can be used to construct biembeddings of
cyclic cycle decompositions of the complete multipartite graph
$K_{\frac{2nk+t}{t}\times t}$ into an orientable surface. In particular, we
construct such biembeddings providing integer globally simple square relative
Heffter arrays for $t=k=3,5,7,9$ and $n\equiv 3 \pmod 4$ and for $k=3$ with
$t=n,2n$, any odd $n$.
