My 37" LG LCD TV (37LH20) has no picture, but the sound still works. I've read a few things online that say it is likely the inverter board? I found a 'power supply/backlight inverter' for it on ebay.

SOURCE: Power supply unit, TV cuts off after 20 seconds

i would recommend ebay, they usually have lots of parts for lcd tvs

SOURCE: 37 inch LG LCD won't turn on due to power failure

Power failures can cause all types of problems, have you firstly checked the fuse in the mains plug? If that is ok then it may have blown an internal fuse in the power supply board, these are there to protect the rest of the circuit from power surges.

SOURCE: LG RZ 37LZ55 37 in HDTV LCD Screen Dead

This is likely due to a failed power supply. Replacing the power supply board should solve the problem.

http://www.reviewcentre.com/reviews110919.html

Cr1spy

SOURCE: LG 42PX4MVH PLASMA TV - Sound but no picture.

I have the same issue with my LG 42PC3DVA and I narrowed it down to the Y SUS board. After a little research there are a few items that are prone to go. The ceramic fuses (as you found), sometimes the capacitors on the Y SUS, Z SUS, and power supply go along with them You can tell because thy will be bulged at the top instead of flat. I replaced a few of my capacitors - they are only $0.89 each so it was MUCH cheaper than a board.

The other thing that is common on the Y SUS in particular are the FETs (also called MOSFET). They can be replaced if you are good with a soldering gun but they are static sensitive so be grounded. They are also a bit tricky to test so you may want to do some research on how to determine if this is the issue... or you can just replace the whole board.

SOURCE: 37'' LG lcd tv, no power

I just want to clear things here.
The stand by light will remain on whenever the tv set is turned on by a button on the side of the tv.

Then use your remote to turn on the tv, but wait as it can take 10secs before you get a picture.