Some insurers will, however, allow you to start building up NCB on a moped, so long as you take out the full policy with them (they won't carry over). A friend of mine did that, built up 2 years on a bike and his first year on a car was by far cheaper.
There's also the pass plus which most insurers dn't give a **** about, but some will make a reduction of you ring them up and tell them you've done it.
Best thing is to just ring about, generally best prices are found online but you can get some deals by ringing up and having a word to an actual person.
There's also the smaller insurance brokers, a mates dad is one who would swear blind that he could beat the price and service of a bigger company such as Churchill everyone me but whenever I asked him for a price it was always dearer.
Other things that can bring down the cost of insurance are safety modifications like roll cages, racing seats with three point harnesses etc.