Two identical piles are formed from finely chopped weeds (mowed and bagged in my lawn mower).  A layer of partially finished compost is added to help speed things up.
7/4/2009
More of the chopped weed material is added to make the piles about 3 ft high and around 5 ft across at the base.
7/4/2009
One pile is left as is while the one on the right is covered with a sheet of bubble wrap folded twice to produce an insulating layer four plies thick.
7/4/2009
The insulated pile is then covered with another large sheet of plastic with the edges held down tight with top soil.  This helps seal off the pile from moisture loss and makes the insulation more effective.
7/4/2009  Whoops, mistake!  Had to take outer cover off and let the pile breath a little.(July 10th)
Updated August 4th: This is a demonstration to show the effects of insulating a compost pile.  The two sample piles shown below were created on July 4th, 2009 with the insulated pile covered by 1:00PM.
By 11:00 PM only the core of the conventional pile was heating up . . . while the plastic near the edge of the folded bubble wrap on the second pile was feeling quite warm indicating that the insulated pile was already heating up even to the very surface of the pile (also indicating heat loss in spite of insulation).
Click on any of the pictures below to enlarge.
Day 1 results:  To be as fair as possible, I read the temp in the insulated pile("H") first and then let the needle fall back to the lower temp while buried in the uninsulated pile("C").  Also note that I kept the thermometer pulled up on pile "H" and buried 6 inches deep on pile "C".
You want to read the heavy black bar which shows just above 150 degrees in pile "H" and would be around 130 degrees or so in pile "C".  That is a 20 degree difference in just one day . . . at 6 inches down in pile "C", and near the surface of pile "H" !!!
Click on 1st pic and then at the top you can click to next.
Insulated pile "H"
Uninsulated pile "C"
Day 2 thru 7 results:  Things leveled off by day 2, in fact the temps in pile C were a little higher for two days.  Both piles have been running between 120(AM) and 130(PM) .  It is obvious that far more activity is taking place in pile H at the surface, and the biggest difference is noticed at the surface closer to the base. Near the perimeter of pile C you have to push in deep to show much of any heat, but pile H shows some heat right up to the surface at any point that it is covered.  I did take off the top layer of plastic on pile H because I was smothering the pile.  It does need to breath some even while utilizing cover to hold moisture and insulation to hold heat.  I plan on building a super insulated pile by winter with an R factor closer to twenty or more.  Got a lot of rain today and therefore pile C added moisture while pile H stayed as is.
After 1 month:
I wouldn't call this test a failure . . . but I need to set things up a little bit different for my next test.  I should have created better piles to start with which would have meant a better mix of materials, and more moisture added to begin the process.
I have two other piles that stay covered and stay very hot and very moist, but they were constructed ideally.  By building these test piles from mostly chopped weeds, they never did much after the first day and cooled down a lot after two weeks.  Ironically, we have had a cool and rainy July, so the uncovered pile has had access to much more water than the covered pile. If not for the test, I would have pulled back the cover during some of the rain, but I didn't.  The rain wasn't enough to help the uncovered pile do much, and the covered pile has stayed moist immediately under the cover but is pretty dry inside.
I don't want to poke around in the piles and "check" them a lot, because that effects what is taking place.  I am going to continue to leave them sit for another month or two and then open them up and check the differences.
In the mean time, I am going to set up another test as well as report more on the two covered piles that are working so well.

What I have learned and thought of so far is this:
 
1. An ideal pile has to be able to drain effectively. In order to get hot and stay hot, a mixture of course and fine material is almost as important as the C N ratio. Materials such as kitchen scraps and even grass clippings need a lot of bulking material such as wood shavings or straw to keep air spaces in your pile. If your pile is slime, you need to add a lot of ruffage.
2. A pile has to be substantially moist at the start. Even a covered and/or insulated pile will loose some moisture since you need to leave enough "gaps" to keep it from completely starving of oxygen. (of course there are ways to add oxygen so that the pile can be completely sealed off).
3. An ideal way to add moisture and some oxygen is by having a urinal drain directly into your pile. This will also add nitrogen as well.
4. Watering a pile, especially an uncovered pile, will cool it off, sometimes more than the bacteria can overcome. One way to get around this would be to water your pile with a hose that has been sitting in the sun. Stop watering when the water begins to feel cool. You could get fancy and have your urine line empty into a solar heated "reserve" which could be then "emptied" into the pile in late afternoon when temps were highest.
5. Covereing and/or insulating your pile does make a huge positive difference, BUT you have to have an ideal pile and ideal conditions from the get go in order to maximize and appreciate the difference it can make.
6. It i difficult to overwater a properly constructed, well draining pile, except for that you may cool it off too much if the water that you add isn't hot.

BTW . . . as it continues to rain, I think I may water my covered pile one time and see if anything happens. I suppose I could even turn it as long as I turn the other one also. ??? I don't want to be perceived as trying to alter the results, but the open pile has received a lot of rain, and if not for the test I would have uncovered the insulated pile during some of these rains.  It's just one test, and more will follow.  I sure wish I would hear from someone who has now covered their pile for the first time.