The counter service burgers are all on wheat bread.
As far as the taste, you just opened up the can of worms. I personally think the buns are so bad (the hot dogs one are wheat as well) that I will never order another hot dog or hamburger at Disney World until they give us a choice between the wheat ones or the normal white ones. I don't care about nutrition, healthy, whatever. I hate the taste of the buns. I asked about getting a white one and was told no. For me, the entire taste of a perfect Disney hot dog is totally ruined.
Now - there will be fifty people who will post they didn't think it was that bad at all. And fifty people who will say they don't like the new bun either.