I have a maytag refrigerator we bought in May of 2012 (warranty expired of course!). The freezer is on top. I have a sheet of ice coating the floor of the freezer on the right hand side. It is freezing and cooling great. It is extremely hot weather right now, but, I'm air conditioned inside. What could be causing this problem?

I have been looking into this because my own Maytag fridge has been acting up. The info I found was that the icing up most frequently occurs when the timer mechanism fails or gets "hung up".
I am just repeating what I recently researched, I have no refrigeration expertise.
The timers are usually identified as 8 1/2 hour or 12 1/2 hour or similar. The unit runs for 12 (or 8) hours, then goes into defrost cycle and a heater comes on to melt the ice buildup off the coils.
If the timer fails for any reason, the unit continues to run with no defrost cycle and the freezer gets colder and colder and ice blocks the air flow over the coils.
I watched a video online which showed a manual override for the timer mechanism. You can turn it slowly with a screwdriver until it clicks to the defrost status.
This is the video which gives good information on how it is supposed to work (you have to sit through a 3+ minute ad before it gets to the nitty gritty tho) but it may give you some ideas about what is going on. The link refers to "no cool refrigerator" but the video addresses the icing up problem causes-- as the ice buildup also contributes to high refrigerator temps. Good luck.
Troubleshooting a No Cool Refrigerator - Part 1 - YouTube (http://www.youtube.com/watch?v=gaRyNiPn26o)

There are two possible reasons: A stopped up drain in freezer, a failure in defrost system. You will need to check the defrost timer before disassembling to see if it will go into defrost and turn on the heater in freezer and advance to turn unit back on. Either way you need to remove the cover (after unplugging unit and thawing out freezer) of freezer coils. Then look at drain and pour warm water down drain or flush with hose nozzle. You also need to check the defrost heater with an ohm meter. Here are the manuals: https://dl.dropboxusercontent.com/u/8010163/Maytag%20Refrigerators.zip