mistersubliminal, I suggest the following steps to isolate the problem.
Is there a pattern to the condition? Disregard the mechanical damage done by dogs and children. That is a G&BI (grin and bear it) condition, to quote Dr. Gary Simone. Circular patterns can indicate a fungus problem. Other patterns, especially if they coincide with wet areas e.g. lawn mower wheel tracks, can also indicate a fungus infection.
Next, sample the earth in the areas of good, bad and indifferent growth. Try to determine whether the soil moisture has anything to do with the visible symptoms.

Do a 2 2 2 flush. 2 oz. of dishwashing liquid in 2 gallons water poured GENTLY over 2 sq. ft. of the lawn. Give the solution time to soak in and watch for anything that crawls, wiggles or leaps to the surface. Capture and identify them in order to ascertain whether you have an insect problem. Grubs are almost a separate nation. They come to the surface, feed on the grass roots and retreat downward in the soil and wait for another time.
If you can pick up any area of your lawn like you would a rug, grubs have done their mischief. You would have to dig deeper (12 to 18 inches) to find them at this time of year, I think. (It is almost a full time job tracking the weather conditions in my neck of the woods). For this activity, a trowel works fine.

If all of these interesting Saturday morning activities do not yield any clues, then you may have a chemical problem, either in the soil or in the grass itself.
The County Extension Service (there is one near you) is the best resource at your availability to help with soil analyses (pH, exchangeable pH, CEC and similar mysterious abbreviations) and plant analyses (N, P, K, Ca, Mg -more but perhaps less mystifying abbrevs.).
I must caution against these cost-bearing analyses unless the results come with practical recommendations for remedial action.

Thank you very much for providing an opportunity for me to get my thoughts together on this fascinating topic.

Thanks for your suggestions, ronalawn82. I have no circular patterns, nor do I think I have grubs, but I will take your advice and see what happens. All the best.

Perhaps your lawn isn't getting enough sun. You also may have some soil compaction in high traffic areas. How many hours of sun per day does your lawn receive during the summer?

-Deerslayer

The lawn is fairly small, probably about 600 square feet. It is an odd shape and gently sloped, with a number of trees around it. I have cut back the trees somewhat to increase lightflow to the grass. I'd say that in the summer different parts of the lawn receive around three hours direct sunlight and another three hours of tree-filtered or dappled light. As for compaction, before the sod went down I loosened the existing soil with a bow rake and added about an inch of new topsoil. I know that doesn't loosen things up like roto-tilling but that is history, now. Towards the end of the summer I spread aroud about a 1/4 - 1/2 inch of Black Kow mushroom compost. When I get home I will do what I remember being called the "screwdriver test" and see how hard the soil is. I welcome further suggestions.

I think that the shade is your primary problem. Here's a quote from the University of Missouri Extension web site:

"Zoysiagrass grows best in full sun and requires at least six to eight hours of sunlight. It will tolerate moderate shade, but the turf will be thinner and less competitive in shady areas."

Here's a link:

Zoysia

-Deerslayer

So if sunlight is the problem, and I were to lay some seed down this Spring, what kind of grass seed would you recommend?

If you live on the SC coastal plain, St. Augustine is a possibility. It has better shade tolerance than Zoysia.

Here's a link to a Clemson web site that has a table that includes shade tolerances for SC grasses:

Clemson

Since I do not have direct experience with SC lawns, I suggest that you go to a local nursery and ask which grass and variety grows best for your conditions.

Also, you may want to consider not having a lawn in the shady area. Other options include ground cover, deck, patio with built-in BBQ, fish pond, shade garden, swing set, etc..

-Deerslayer

BTW, St. Augustine is normally propagated by sod, plugs, or sprigs. Seeding isn't a practical option.

-Deerslayer

Thanks a lot, Deerslayer. You've been very helpful.

How about that the ground was crappy to begin with. You stated that the lawn was a mix between weeds and dirt...I don't know about you but where I come from here in north GA weeds will grow just about anywhere. If a weed isn't going to grow somewhere I'm pretty sure a lawn isn't going to magically take root either. Try putting down some compost. If it's bad enough and you don't mind tearing it up a little mix it in. Otherwise just use it as a top dress. Keep the organics going and see if you can find someone near you that might have a compost tea that they can apply. The idea is to treat the soil, if the soil is inhabbitable then nothing will grow. You need to get the microorganisms back in the soil and thriving and then you shouldn't have any issues. Build your soil up from there, maybe put down a layer of compost one or two times this year to try to build the soil back up some. This should make a difference. You can find some more products at some websites around that will tell you what to put down. Basically stay away from any synthetic chemicals as these usually harm the microorganisms that your are trying to restore.
Also make sure that the compost is a finished product. It should be a dark brownish color and not smell bad. This should have plenty of beneficial nutrients and organisms that your lawn sounds like it desperately needs.
Healthy soil = healthy plants.

Good luck,