What is the easiest way to draw a long 3D tube that follows a 3D path.

I have a collection of 3D points that represent a route through my world and I want to highlight it. If I draw it as a THREE.Line it comes out 1 pixel wide which is impossible to see, so I want it as a 3D solid.

I tried extruding a shape along a path but that has a couple of issues - the path is defined in 2D and the shape (which is also 2D) doesn't rotate as it is extruded so I get thin points in my solid.