I have an shrub type rose in my garden, the sort with lots of thorns and wide pink scented flowers, not sure of the exact name. The problem is that it's producing lots huge long non flowering shoots from it's roots. I believe these are called runners.

Is there any solution apart from cutting them back constantly or digging up the whole bush?

I think you mean suckers. I have a problem like this too as I inherited some overgrown bushes and cut them back best I could. Trouble is I cant tell the suckers from the real growth until they grow a bit. There are lots of congested stems and its really hard to tell which is from the root and which not.

So Im watching this space too.

Yes you are right, they're called suckers.

I was wondering why I couldn't Google the problem, but with the correct word I now see all this :)

http://www.google.co.uk/search?hl=en&q=rose+suckers&meta=

Thanks for the link. I think my problem is the previous rose bush owner cut the suckers back but didnt completely remove them. So now I have suckers growing on the cut back suckers IYSWIM.

Sounds like a lot of hard work.