Steve, I'm not going to get involved in the current discussion. You've got Marshall and others to handle your details. But there are a couple of things said in this thread I'd like to address from my experience.
Obviously there are different quality and kinds of landscape cloth. My experience with the woven kind that I've used is not with weeds growing through it but with Bermuda grass growing through it. The Bermuda grass sends out runners that run all under that landscape cloth. It does the same if I use newspapers or cardboard covered with mulch too but the Bermuda does not send shoots up through them. Those runners can run for several feet. Every joint is a place roots can form and shoots can come up. I have not experienced "weeds" growing through it, just that Bermuda grass.
With the landscape cloth I use being woven, roots will get up in the weave and maybe run 6" to 10" embedded in that landscape cloth, then send a shoot up through it to grow. Those are a pain to pull out. Those roots are like they are woven into the fabric. Here my only problem with this is the Bermuda grass but from what Journey said, it sounds like quackgrass is similar.
The other thing is about things sprouting in the mulch. I use wood chips on top of landscape cloth in my "ornamental" areas. These are just wood chips I get from a nearby city that chips the branches that people drop off or their street crews clean from the roadside. You might find oak, cedar, or some other good stuff in there but a lot of it is from what I call trash trees. Also, very little of it is heartwood. It's mainly new growth that rots, composts, decomposed (whatever you want to call it) very quickly. I have to remove and replace it every year or it rots and provides a good medium for seeds to sprout and grow. Bermuda grass runners also run in here, root and send up stems and leaves. These old partly rotted wood chips is usually what I mulch my tomatoes with, spreading them over two to four layers of newspaper.
Most of the dirt falls out of them during handling. When I replace the wood chips is also when I pull up the landscape cloth to get the rotted "dirt" off of it (which I try to save and use as compost) and pull the Bermuda roots out from under it and try to remove those roots that have grown into it.
My biggest problem area is under that ground-hugging spreading juniper with sharp needles and where the Bermuda grass has grown up through the X where you planted it. That's painful and I'm not real efficient in getting all the Bermuda grass roots out of there.
At least here, I can't spread the landscaping cloth, cover it with wood chips and walk away for years. It does greatly help in keeping the weeds and even the Bermuda under control in those areas, but it requires annual maintenance.