  The flip graph of triangulations has as vertices all triangulations of a
convex $n$-gon, and an edge between any two triangulations that differ in
exactly one edge. An $r$-rainbow cycle in this graph is a cycle in which every
inner edge of the triangulation appears exactly $r$ times. This notion of a
rainbow cycle extends in a natural way to other flip graphs. In this paper we
investigate the existence of $r$-rainbow cycles for three different flip graphs
on classes of geometric objects: the aforementioned flip graph of
triangulations of a convex $n$-gon, the flip graph of plane trees on an
arbitrary set of $n$ points, and the flip graph of non-crossing perfect
matchings on a set of $n$ points in convex position. In addition, we consider
two flip graphs on classes of non-geometric objects: the flip graph of
permutations of $\{1,2,\dots,n\}$ and the flip graph of $k$-element subsets of
$\{1,2,\dots,n\}$. In each of the five settings, we prove the existence and
non-existence of rainbow cycles for different values of $r$, $n$ and~$k$.
