Unsetting (or even changing the value of) MAILCHECK didn't stop the message printed when I open a terminal tab/window. I eventually stumbled on an answer on superuser that enables me to still see this mail when it's convenient, but skip the constant nag message.
In short, create a
~/.forward file and, on a single line, the path of a file you'd like to receive mail in. Later, you can still elect to read the mail with
mail -f <new_mail_file>.
If you have more advanced needs, it looks like forward can also specify a script by setting the line to
"|script_path args". I haven't tried this, but it should make it pretty simple to route messages based on the script that produced them, convert/compact them into a simple summary logline, etc.