I seem to remember a few years back there was a program run by one
of the airlines where for a fee of around $1500 or so you would go and
train for a weekend on a Boeing 737 and at the end you would actually
get to fly one for half an hour or so.

I don't think you got to fly a real one, just the simulator. UAL used
to offer it to anyone. If you check the flight simulation groups,
you'll find that their yearly get-togethers almost always include
optional signup for an airline simulator... UAL was last year, I think.
Also...

http://www.b737.com/
http://www.smartt.com/~simflts/

I know there are more in England that offer this to anyone. Try some
Googling.

Best, Kev