On the sharpusa.com web site I could not find a manual for the 37D62U, but looked at the closest I could find, the 42D62U.
From your description I was wondering if maybe the TV didn't support HDCP, but it does. I couldn't find what version of HDCP & HDMI so can't confirm compatibility.
The manual does have a table of supported video modes (PC format compatibility chart). As a last resort you may have to fiddle with xorg.conf and these frequencies. You really shouldn't if you are using 1080p.
One interesting note under "Input Signal (for PC format)" where it says you may need to select between 1024g768 and 1360g768 under MENU, Setup, Input Signal as the TV can not auto-distinguish between them.
You might want to try different "View Mode (for PC format)" to see if you can find a usable mode.
You probably have already tried this, but maybe just try repositioning the image (MENU, Setup, Position). In the same line of thought, there should be a Troubleshooting appendix with a "Picture is cut off/with sidebar screen" problem.
If you haven't already, try a different input source to the TV (eliminate the possibility that the problem is with the TV).
You might want to try plugging in a VGA monitor to the K9NGM3's VGA port in addition to having the TV hooked to the HDMI port. The AV Setup Wizard will display on the VGA monitor until you get to the point of trying the display modes. In truth, this is the way I always setup a system.
HTH,
Roy