Question: Is there any way to coax Gmail into pushing new-mail notifications into my Linux machine, without using a full-on graphical mail client like Thunderbird?

**edit**: Thanks for all the responses, but (unless I'm mistaken) these applications all *poll*, none of them receive notifications *pushed* from GMail.

Also, I'd prefer a console-based program, as this will be running on a headless server.

Answer: 
There is a program called checkgmail 

(<http://checkgmail.sourceforge.net/>)

that will sit in your gnome taskbar. I know there is a plasma widget for KDE

(<http://www.kde-look.org/content/show.php/gmail-plasmoid?content=101229>)

that does something similar, although from what I've read, it might be a little broken.

The program Docky2 has a pretty good gmail notifier built in if you use a compositing window manager like compiz or metacity.

(<https://launchpad.net/docky>)

I don't know of any terminal based checkers if you don't have an X11 at all.
Gmail doesn't push. You can receive an atom feed:  

<http://code.google.com/apis/gdata/faq.html#GmailAtomFeed>