Recovering doors of Florence Knoll 116 sideboard  

Horrible!
I believe this is the stuff, but I might be wrong:
Close weave rattan webbing:
http://hhperkins.com/index.php/cPath/12_30_47

Hessian is burlap--it's not that and it's not woven rattan, either. I believe it's just seagrass wallcovering. There are man different weaves of seagrass wall covering and I think this one is called "Akiko", maybe...?? anyway, here are some examples plus a photo of an original Knoll door. The slubs (bumps where the fiber is thicker) are an important characteristic.
This stuff is widely availalbe. Don't order it online, though--just go to a wallpaper store and look at the sample books. There are usually a few that are just seagrasses. It comes in many different natural colors and you might be able to find with the golden color of old seagrass if you're lucky. The quality ranges from cheap to very high end. I have some that I got on Ebay that retailed for over $300 a roll at the time, maybe $400?? It was really great stuff. I papered a room in my old house with it and I would have taken the paper off and rehung it in my present place if I could have! It was gorgeous.

It is called Pandanus cloth, one of the ealiest textiles imported (from the Philippines) by Knoll in the 1950's (See 'Knoll Textile 1945-2010' book).
It is not in the current Knoll Textiles line right now but it was offered for a while. (I took home a 2 yard sample being discarded from our materials library a few years ago). Your best best is to look around from online sellers.
The same material is also used on sliding doors on cabinets for Dunbar, Nakashima studio, McCobb Planner Group and others.

Request several samples of woven mat/straw materials from different manufacturers like this from Ralph Lauren and compare to see if it matches the size and density of the weave of what you have on your doors.
http://www.ralphlaurenhome.com/products/wallcovering/item.aspx?haid=58&c...