Nah. It predates Jackson and shows up in video games and even in how they talk in some D&D or other gaming-related novels. Warcraft video games had it, back in the 90s (and those stemmed in part from Warhammer, which I think had it), and there's a Poul Anderson novel from the 1960s that has it as well. Some people have speculated that the idea of the scottish engineer-type ala "Scotty" from Star Trek is the root of it.
My hypothesis on the connection between Dwarves and Scottish accents: There's supposed to be a gene for 'engineering talent' and such (I'm not going to take sides on that) which is reputed to be more common amongst people of Scottish ancestry. Dwarves are generally supposed to have innate skill when it comes to making things (engineering), especially things made of metal. So the "logical" conclusion was that Dwarves would sound like Scots...
(The cliche of the Scottish engineer is far older than Scotty from Star Trek. Could have something to do with a Scot inventing the steam engine, my twin says.)