The screen goes black but the sound is fine. If I unplug the cabe to the 6870C-0193A board, the TV will work for a few days and then it goes black again.

Intermittent problems are the hardest to solve.
You did not give a make and model but based on the part you gave I think it is a
Toshiba 37AV500U or a Toshiba 37AV502U or a Visio VW37L HDTV40A or similar model.

The part number you gave is a T-CON board which drives the LCD screen.
This board sell on the internet for between $30 to $75.
eBay
http://tvparts4less.com/index.php?main_page=product_info&cPath=286_321&products_id=5407
http://alltechdepot.com/vizio-v037l-hdtv15a-lcd-controller-vsc-pt6870c0193a-p-3085.html
etc

Normally if the screen goes black it is an inverter/backlight problem. Disconnect/Connect it in case the connection is tarnished.
Since you are working inside the set I assume you know something about electronics repair and the warranty has expired.
The next time it goes black, place a bright flashlight against the screen and see if you can see and image around the edge of the flashlight. If so the inverter is not working.
Try gently pushing on the cable to the board and see if it affects the screen. Bad T-CON/connection usually causes lines on the screen or parts of the screen display to be missing.

You can replace the most likely board and see what happens. Even if you replace both boards it will still be less expensive than taking it to a shop.

Clean the connectors for that cable and the connectors for the board that should do the job. good luck.