Inkscape's default option to Save As > Desktop Cutting Plotter (AutoCAD DXF R14) (*.dxf) will result in an empty AutoCAD drawing. It took me a while to figure how to fix it, but thanks to MrGoon, I finally have a straightforward solution.
1. Open SVG file in Inkscape
For the sake of exercising, you may download a free SVG file that I used here.
2. File > Export Bitmap
3. In Export Bitmap window, specify a high resolution, such as 600dpi. This will create a PNG file.
4. Open the PNG file in GIMP
Steps 5 to 8 essentially erased everything but the black outlines.
6. Invert selection
7. Clear selection
8. Select None
9. Copy and paste into a new Inkscape file (Change document orientation and reposition pasted image if desired.)
10. Path > Trace Bitmap
11. File > Save As .dxf
Et voila! When you open the .dxf file, you will see the black outlines appear in perfect "health".