This is a process that took me a while to figure out. Here's the step-by-step to make it easy.
Note: This will only work for files made in Inventor Pro, I still have not figured out how to get files from the student version into Illustrator.
Now, for this next step you have to use a program called Autodesk Trueview. This is a free program that you can get here from Autodesk's site. Open the .dwg file you just exported in Trueview, and hit DWG Convert, a button in the top left of the window. Select the file you want to convert, and set the conversion settings to 2004. For some reason, Illustrator just can't open .dwg files that aren't in this format. If you're going to use the file in VCarve, this is the last step of conversion.