Home Made Mince - What cuts to use for great mince beef, lamb etc

I have a hand mincer and want to make my own mince without all the nasty stuff that seems to be in ready prepared mince.

What cuts of meat are appropriate to make good quality mince that melts in the mouth after only being cooked for a short while at reasonable prices?

For example my local supermarket has beef roasting joints at half price - would this be any good?