Graphs in which every minimal separator is an independent set
Your graphs have been characterized by this paper http://arxiv.org/pdf/1103.2913.pdf. 

Edit: In the paper above it is proved that graphs in which every minimal separator is an independent set are exactly those containing no cycle with exactly one chord.

Graphs containing no cycle with exactly one chord have been studied in depth by 
Trotignon and Vuskovic, A Structure Theorem for Graphs with No Cycle with a Unique Chord and Its Consequences,  J. Graph Theory 63 (2010) 31-67 DOI. 
As a result of this paper, these graphs can be recognized in polynomial time. (However, this paper did not point out the connection to independent minimal separators!)

Edit (17 Sept 2013): Very recently (see here), 
Terry Mckee describes all graphs in which every minimal vertex separator is a clique or an independent set. It turns out that these are the ''edge sums'' of chordal graphs and graphs in which every minimal vertex separator is an independent set.