Burley needs an image lift, if you ask me. Sure, some of those OTCs get blasted for using cheap Burleys, but more likely, the problem is fast processing and drenching in goop. And some are not even half bad, if you like that sort of thing.
I have been experimenting with Burley as the majority of a smoke and I like it because the absence of excessive sugar means that the smoke is light and pleasant, with a rich flavor that I personally enjoy. Generally, a blend that is 70-85% Burley -- mixed white and dark -- works well as a canvas on which to splash condimental tobaccos, including Virginia as a sweetener.
In addition, it is impossible to tire of the eternal blend of 3:1 or 3:2 Burley to Virginia, which produces a warm caramel-honey flavor with bready undertones. Add in some Perique to give depth and tang to the sweetness. The Va/Bur/Per may be one of my favorite types of blend, if the Virginia portion is kept low.
As far as straight Burley goes, try Cube Cut Burley. You can smoke this all day for the flavor equivalent of a freshly mowed field. Or Cotton Boll Twist, which is not the monster that some have portrayed it to be, but an enjoyable smoke with flavors of nuts and hay dried in the August sun. MSO489 mentions La Brumeuse, which is really excellent once you get over how dry it is, and some of us are fans of Five Brothers, a dry shag-cut Burley which is great on its own but venerable in its ability to shore up other blends.
Right now I am burning down a bowl of 60% Five Brothers and 40% Early Morning Pipe, a combination that has fascinated me recently. I taste more of the salty Balkan blend that is the EMP, and it gains a roasted and sturdier flavor from the Burley.