I tend to favor fast and total eradication but I will share what little wisdom I have in case you are of the more cautious sort.
A lot of evergreen shrubs are shallow rooted so be careful with your shovel around those. The burning bush roots can get tangled with other plants' roots. You might want to test and see if the burning bush roots are tangled with your natives by starting to pry them out and see if you get movement or ripples by your natives. Privets are supposedly easy to pop out with a shovel although I haven't done much with privets so I can't say this from personal observation. A Weed Wrench is good for jacking woody plants out of the ground. The burning bush roots will be more extensive than the privet roots. The only potential danger to painting the stumps with Round Up is if the roots of your targets have grafted to roots of your natives. I don't know if privets or burning bush roots ever graft to other plants' roots so research is needed on that point. To be much safer you could cut them all back now and remove every other or every third shrub (roots and all) a few times a year over a period of a few years. Repeated cutting will off burning bush - I don't know about privet but I suspect that there is only so much even the toughest invasive can take. Some seed from your targets is probably in the ground now so a few new sprouts is possible but will be much easier to handle. Regenerating root fragments can be a problem with burning bush but they do die off sooner or later.

Bob64,
Thanks for the suggestions. Is there one time of year that is better than another for removing shrubs? Should I cut them back now and haul out the roots in the spring? Or do every thing now? Or doesn't it make much difference?
Martha

If your natives have gone dormant already, the disturbance will be less likely to damage them. For me, the reality is that I do things whenever I can find the time. You could even just girdle your invasives now if you need them to block wind or provide some sort of cover or to hold soil. If you dig up now, make sure to tamp down the soil again and mulch. I would not try to uproot or dig after the ground is frozen.

If I was trying to remove the shrubs I'd cut them off near the ground then keep cutting any sprouts that appear. I'd go out and cut them now - my approach is apparently the same as bob64's in that I remove unwanted invasive plants when I have the time regardless of the time of year. If you cut them now I expect that the shrubs will resprout vigorously in the spring. At that time go out and cut off any sprouts trying to completely remove them or at least inflict as much damage as you can. Try not to let the sprouts get more than a foot tall because they will be creating new energy reserves for the shrub that will support even more sprouting. I think you'd be OK cutting sprouts once a month. By the end of the summer, or perhaps next summer, you'll notice much less sprouting, indicating that the cut stumps are running out of stored energy needed to create new shoots. Eventually the stumps will die.

Alternately you could cut now, then when sprouts appear spray with Roundup or similar herbicide. It might take more than one round of herbicide to completely kill the stumps - if new sprouts appear then you need to spray again. There is a danger that herbicides could injure the nearby plants that you want to keep, so if you use herbicide try to use the minimum amount needed.

Once the stumps are dead they will weaken and rot and you can break them off at ground level, leaving the roots to rot in the ground.

If you cut down to stumps also cut a little "x" or make hash marks on the top of the stump. This increases the likelihood that your invasives will rot or sucumb to something nasty. I hate digging but I will pull or pry out smaller stuff. You can paint on the Round Up if spraying is too dangerous.

Thank-you all for your suggestions. I think I'll tackle this project in the spring when the bushes are actively growing.
Martha

Some (maybe all?) types of privet can sprout from root fragments. If you dig out the stump, you will have dozens of plants where you used to have only one. The only way to reliably kill your privet is to spray herbicide on the foliage or paint a freshly cut stump.

CV

i prefer shaped explosive charges and small arms fire myself....

I know this post started four years ago but I'm also trying to remove a really big bush from my backyard. I don't know the name but it looks like the agapanthus family. I wish I had tried when I first moved in six years ago. But I'm not going to give up...I need some privacy on that side of the yard and this is the only plant that has to come out. I want something tall like your privets...I have a few on the other side of the yard and they make a great hedge. Thanks for all the great suggestions...it helps.

I need to remove the shrubs by chemicals not unrooting them, is there some kinda of chemical that will kill grass and shrubs so i can replace them with sea shore stones?

We have tons of huge, well established azaleas on our property (about 2 acres)... along with probably 50+ camelia bushes. We are thinking of thinning out the yard to reduce the stress on the grass and soil. Hubs says to chainsaw down the ones we don't want... keep mowing over them every time sprouts come up and everyting will be fine. I say that isn't good enough and won't do the job.

If we chainsaw... can we put some sort of root killer on the stumps? Could this harm other bushes near by ( which would be other azaleas or a neighborly camellia)...

I love fast and easy... I just don't want the yard to be littered with hundreds of "stumps" that become a nuisance for the lawn mower.

I guess I can't have my cake and eat it too... right?

Do you have access to a truck and winch? I have often fantasized about getting someone to pull these buggers out of the ground. Quick and easy.

However, since I haven't had that, I have dug many shrubs and some are easier to dig out than others. Forsythia and bush honeysuckle are relatively easy to dig. Buckthorn I would never try to dig, their roots are like concrete. Likewise Norway maple. I've dug out large Burning bushes (30-40 years old). They are tough, but it can be done. I was younger and more vigorous then though lol!

My experience with transplanting Rhodies and Azaleas is that they have shallow roots. Pretty easy to dig.

These days, when removing invasive shrubs, I usually cut and poison with a concentrated stump/brush killer (Brush b gone). Sometimes I cover the stump with a pile of organics to encourage rotting. When a stump remains moist it rots out much faster. This can also discourage resprouting, as no light reaches the plant. The piles are uglier than stumps littering the yard though!

Terrene, I was going to say the same thing - yank with truck but do it smoothly, if that makes sense.

Also wanted to say to call "call before you dig" before you dig. Besides being fun to say, it alleviates you from responsibility if you are not correctly notified of any buried pipes or other utilities.

Our house is part of a government buyout program along with the others on my street. The houses will be removed and I plan to take a good part of my garden with me. So far I have dug up and repotted a fig tree and a pomagranate, but I am having trouble with my Japanese maple and my camillia bush. I have dug a shallow trench around each and will continue to dig deeper and deeper until I feel that I can remove the bushes without destroying any more of the roots than necessary. Do you have any better suggestions? This is proving to be a lot more difficult that I had imagined. I am only doing this because I really love these plants and have babied them for years.