What is the name for the flat attire worn over the shoulders at academic graduations?

Solution 1:

Graduation stole or academic stole:

A stole takes the form of a cloth scarf-like garment worn over the shoulders adorned with the awarding Society's colours and/or insignia. Though not a part of officially recognised American Council on Education's Academic Costume Code, it has become common as part of the graduation attire at many high schools, colleges, and universities.

Here's a photo from the same page with the graduation stole highlighted:

^{By Valedictorian_Joint-Chiefs-of-Staff.jpg: Chairman of the Joint Chiefs of Staffderivative work: Themightyquill - This file was derived from  Valedictorian Joint-Chiefs-of-Staff.jpg:, CC BY 2.0, https://commons.wikimedia.org/w/index.php?curid=20548944}

[Wikipedia]

Apparently, the term graduation sash is common too:

The colors of graduation hoods, cap tassels and stoles, also called graduation sashes, are governed by traditions from the Middle Ages, reports the American Council on Education, or ACE. [ehow.com]

However, a sash typically goes diagonally across the body, like what pageant queens wear (see WBT's comment below). Sticking to stole is the safest thing to do.