Thank you everyone for your replies. It is indeed odd! I am on a learning curve with the 5DIII, but have a 50D and 550D, so am familiar with EOS utility, DPP and Adobe Bridge. I did wonder if there was something special about the RAW files on the 5DIII.
I don't have a card reader (yet!), but I have tried using each of the cards separately, and reformatting them and starting afresh. The red herring was when it downloaded fine, but that was because of the factory default settings - it's set to jpeg Large.
With raw files, the EOS utility just hangs until I have to force quit it, and adobe bridge hangs and eventually says it can't download them (I can't remember the reason it gives now, but at one point it said it was possibly a low battery on the camera).
I suspected the lead (as did the Canon guy), but I successfully loaded raw files off the 50D with it, so that discounts that.
I did manage to get some RAW files off if I tried one at a time. As soon as I tried more, it hung. Luckily I've just been playing trying to familiarise myself with the camera, so nothing important is lost.
My daughter tried downloading them with iPhoto on her Mac. That hung for a while, and then she clicked to cancel, as it said there was a problem… but it did then download them…. strange!
I think I'll just have to be patient and await Canon's feedback
I will look into getting a card reader and reconsider my workflow once I'm up and running. I don't have Lightroom yet, but lots of people seem to recommend it.
At least I know it is just me :-/
I'm very impressed with the photos I've seen on here. Thanks again for the replies.