I used a cheapy HDMI cable from the HDMI out of my laptop when I tried this on a neighbours 4K telly, only 30Hz but sat close up in front of a 50" screen is super immersive in FSX.
I wonder if there are dual-link DVI to HDMI adaptors, or whether they are single link?
It would also help if the OP mentioned make of TV and the GFX card in use. My Intel HD work PC couldn't drive a 1600p monitor, so no chance with a 4K one,.I had to put a cheap discrete GFX card in.