Dig them up, complete with roots and bulbs.

Not normally that hard to dig up, but you will want to keep an eye out or volunteer seedlings, and pull them before they become too established. A few species also send out rhizomes, and they may take a little extra work to control, especially if they are coming from a neighbors, or an area let run wild.

The bulbs on the ones I want rid of seem to be ten feet down (only a slight exageration). I could never dig them out but seem to have gained some sort of control by pulling out the growing stems. There are not as many as there once were.

Some of us would be delighted to be able to grow alstroemerias..... :-(

I had to dig out a large clump in someone's garden 2 years ago - what a nightmare, took me nearly 2 hours to get it out, and you're right Bulbaholic, deep roots and impenetrable seeming 'pad' of fibrous roots.

The fibrous roots can be let lie, as long as they are no longer connected to a shoot-producing crown. I'm beginning to wonder if we are talking about the same thing--the Alstroemerias I'm familiar with grow well in Mediterranean climates, and have only minimal frost hardiness. 'Dr. Salter's Hybrids', etc.

The alstroemerias grown here are frost hardy.
I grow about 20 of them and they come through hard winters quite well.

The aforementioned roots are well known and i guess that's what makes them such tough plants !!!!!

The grower i know lives in the Worcester area and he usually doesn't protect overwinter and his temps can get quite low.

That may account for the deep rhizomes, then! Chances are, those types won't grow here, as we are almost too hot even for the more Mediterranean types we grow here.

If these get too hot they won't flower as profusely, they'll just produce loads of foliage.

Our heat (46º+ C) tends to kill the less adapted varieties. The better adapted ones simply go dormant in summer.