I'm very organised, just not in any conventional sense.
Rather than attacking it in order of priority, though, I work from left to right, dealing with each item as I get to it. This is a really bad idea for two obvious reasons:
1. The order things happen to lying in on my desk is usually not the best order in which to deal with them.
2. It doesn't take into account the emails, digital files and web-based tasks which make up most of my work these days.
But the need is so strong that, rather than try to ignore it and use an order-of-importance list like a rational person, I've now harnessed it and am using my compulsion as a form of motivation.
I write every individual thing I need to do on its own sheet of scrap paper - we're a newsroom, there's an endless supply of the stuff - and then sort the papers, plus the post-its and beer mats and newspaper clippings, in order of priority. Then the lot goes, in order, on the far left of my desk. Then I have to deal with them. Like a vampire counting rice, I can't move on until I've cleared the pile.
I'm currently training a cat (long story) and it's the same principle. You can't just punish the behaviour you don't want, you have to work with it and find ways to slowly turn it into the behaviour you do.