When the monitor goes out, does the sound still continue to play? If it's a silent video, try playing some music in the background on the PC, such as list of MP3 files or Internet radio.
If the sound stops when the monitor goes out, then the issue is happening at the PC end. For example, it could be something such as a graphics driver issue when using that monitor. In this case try updating the graphics card driver.
If the sound continues, leave the PC running and swap the monitors to see if the video shows up on the old monitor. If the old monitor shows a picture after the swap, then it's likely a fault with the new monitor or its cable. In this case, try a different cable to the monitor if it can be disconnected or try a different connection method temporarily (e.g. VGA instead of DVI and repeat the test.)
Finally, if the sound continues, but swapping with the old monitor while the PC is running still does not show a picture, then it's probably the a graphics card or an issue with its driver.