{ An edge $e$ in a matching covered graph $G$ is {\em removable} if $G-e$ is matching covered, which was introduced by Lovász 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 Lovász'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 $Δ-2$ removable edges, where $Δ$ 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 $G$.