It looks like the designer made a square with rounded corners, skewed it 36 degrees, rotated it 45 degrees, added a version with a big stroke underneath, knocked the smaller one out of the bigger one, copied it and moved the copy below, then added the two extra connectors, combines all the paths, and then tweaked all of the curved connections using circles, the pathfinder, and the pen tool.

The basic structure can be easily made by drawing a regular hexagon (points aligned to the vertical), drawing lines between the points and removing the sides, but I see what you mean, getting the proper radii and fillets in the joints of the strokes will be tricky.

http://www.learnit2.com/tutorial%20023/index.html

You can do "live" rounded corners with Effects > Stylize > Rounded corners. Although you'd want to adjust them manually at the end of the process, this could be good for quick mockups.

Here's a tutorial: http://www.kadavy.net/blog/archive/2004/12/using_adobe_ill.php

Wow thanks for all the great feedback.