My Polaroid Flat screen has no picture, but still has sound

No picture can be caused by dead T-CON board, main board, 24vdc power supply not supplying power to the backlights inverter board, bad backlights inverter board. So inspect for bad caps fisrt and also try the instruction below.
First, look at the screen in darkened room to see if you can see any slight glow from the screen, if it does, then the problem can be the T-CON (not getting the 12vdc) or the main logic board not feeding the signals to the T-CON board.
if not, next place to check will be the power supply module, make sure it has 24vdc which is required by the backlight inverter to work, the reason for not having 24vdc can be from bad DC filter caps (leaking/bulging top, high ESR) or short circuits in the inverter board which will cause the 24vdc supply to go into shutdown mode, so you may have to disconnect the power cable that feed the inverter board to see if the 24vdc will work without the inverter board hookup, you will need to apply about 3~4 amp of load to the 24vdc to make sure it can handle that. There are fuses on the backlight inverter boards also. So there is some troubleshooting you will have to do to find out the causes of the problem you are having.

If you are going to DIY and have proper tools, electronics knowledge, and know safety precaution then please read on:

Basic LCD monitor and TV troubleshooting guide:
http://www.fixya.com/support/r5093881-lcd_flat_panel_tv_troubleshooting_guide
http://www.fixya.com/support/r6150077-basic_lcd_monitors_troubleshooting
http://www.fixya.com/support/r7406380-tv_training_manuals
Failed TV and Monitors pictures: http://s807.photobucket.com/albums/yy352/budm
Learn about bad caps: http://en.wikipedia.org/wiki/Capacitor_plague
Parts: www.shopjimmy.com
Capacitors kit: http://lcdalternatives.auctivacommerce.com/Default.aspx he can make you a set of caps.
Or www.digikey.com, just make sure to use caps with low ESR, 105c, high ripple current and long life rating such as PANASONIC FM or FC series.
Please post back what you find.

There are many, many bad capacitors in the signal board. You must use an ESR meter to locate and replace these. There will also be two bad caps in the power supply, find and replace those two. The signal board will have 15 or eighteen bad caps. If you reassemble your TV and it howls when you turn it on, you have missed a few caps. Take your time, careful not to pull the printed traces off the board or you face destroying this little jewel. Thanks for asking and show a few hands of support.