GL_LINES isn't doing it for me. I need thick antialiased lines, which work fine on my Radeon but not on GF4MX. I also want mitered joins.

So, the GL FAQ says:

"Another solution is for the application to handle the capping and mitering. Instead of rendering lines, the application needs to render face-on polygons. The application will need to perform the necessary math to calculate the vertex locations to provide the desired capping and joining styles."

Sounds like I need to calculate the vector perpendicular to each line join, offset from the line center, and draw a tristrip.

This seems like it ought to be a well-solved problem, so can anyone point me towards code?

Assuming you've drawn your quads to represent the body of the line. I figure you'd have two lines that share one point on their quads. So you could just use a triangle fan with the shared point as the center. Then just do a simple parameterized circle calculation to get the curve to connect the two points that are not shared.

Vague, but I hope this points you in the right direction.