Go with the story order, if only to be different from the rabid 20- and 30-something fans. You will bring something new to any debate about the films if you watch them in that order; they’re so much a part of geek culture that the discussion is sure to come up, and your perspective will be of interest to at least some of the fans. You may be branded a heretic, but you’ll definitely have something original to say.
There will almost certainly be some (jar-) jarring inconsistencies between Episodes III and IV. I would not be surprised if Episode III features a scene in which Obi-Wan witnesses Darth Vader (or Darth someone) betraying and murdering Anakin Skywalker, or perhaps even vowing to “destroy any trace that Anakin ever existed,” which would paint the initial scenes of Episode IV in a very different light than the one we’ve all come to know and love. Better? Nobody knows yet!
Go ahead and watch them in story order. Realize that the first three are being made in the tradition of Episode VI, which is to say they are stained by the director’s realization that a movie can be a blockbuster and a vehicle for toys without being well-directed or well-acted. Episode VI stands on the inertia of the characters introduced in IV and V, and Episodes I and II also lean heavily on what will be revealed later.