Each point interval along the curve (you decide the interval) is a 3D point in space. If you take 2 points you can define a line segment. That line segment could have any number of points around it in a circle perpendicular to the line segment (you decide the number of points). Those points would be one or two vertices of a triangle primitive. The next line segment, using the same idea would be the other side making a set of complete triangles. Repeat this as long as the curve goes and render. You will need to make sure of the vertex ordering so it renders with faces toward the camera; or you could tell DX to cull none.
If you need code, I can help, but not till I'm home.