I am using Branch Specifier option of Jenkins Git plugin (v2.0) to run build on specific branch, e.g. 1.4.
${GIT_BRANCH} in this case conta
While it's true, as dtabuenc says in a comment to Jan Včelák's answer, that shell syntax isn't going to work on token macros directly, if you are trying to get at the branch name in an Execute Shell build step (or similar), you can still use the same idea.
TEMP_GIT_BRANCH=${GIT_BRANCH}
MY_GIT_BRANCH="${TEMP_GIT_BRANCH#*/}"
works fine (and that's valid POSIX, no bash required).