Building on larsk's answer and VonC's answer, you can create a git_ssh.sh script such as:
#!/bin/sh
# Workaround: GIT_SSH_COMMAND isn't supported by Git < 2.3
exec ${GIT_SSH_COMMAND:-ssh} "$@"
Then invoke your git command like this:
export GIT_SSH_COMMAND="/usr/bin/ssh -o StrictHostKeyChecking=no -i /home/me/my_private_key"
export GIT_SSH=path/to/git_ssh.sh
git ...
This is how it works:
In Git v2.3+ $GIT_SSH_COMMAND takes precedence  over $GIT_SSH, but older versions don't respect $GIT_SSH_COMMAND at all.
$GIT_SSH can hold only a path to the ssh command on the system. It can't pass extra command line arguments to that command, so how can we pass extra arguments to ssh?
A workaround is to create a script that includes the ssh command and its extra arguments. This is exactly what the git_ssh.sh is all about: Since we already set $GIT_SSH_COMMAND to be /usr/bin/ssh -o StrictHostKeyChecking=no -i /home/me/my_private_key, it is exactly what we need to exec, and the "$@" is here to pass the arguments passed to git_ssh.sh by Git itself to the $GIT_SSH_COMMAND.
The ${...:-ssh} part, while not strictly needed is a nice touch that will make $GIT_SSH_COMMAND default to the ssh command, and thus setting GIT_SSH=git_ssh.sh will not break a normal git execution.
As added value, this script is totally ignored by Git v2.3+, and the $GIT_SSH_COMMAND is used directly in this case.