  { An edge $e$ in a matching covered graph $G$ is {\em removable} if $G-e$ is
matching covered, which was introduced by Lov\'asz and Plummer in connection
with ear decompositions of matching covered graphs. A {\it brick}} is a
non-bipartite matching covered graph without non-trivial tight cuts. The
importance of bricks stems from the fact that they are building blocks of
matching covered graphs. Improving Lov\'asz's result, Carvalho et al. [Ear
decompositions of matching covered graphs, {\em Combinatorica}, 19(2):151-174,
1999] showed that each brick other than $K_4$ and $\overline{C_6}$ has
$\Delta-2$ removable edges, where $\Delta$ is the maximum degree of $G$. In
this paper, we show that every cubic brick $G$ other than $K_4$ and
$\overline{C_6}$ has a matching of size at least $|V(G)|/8$, each edge of which
is removable in $G$.
