Since you are keep most equipment under the TV on the same wall, I am assuming the max length of the cables will be around 10 feet, if not less.
Just run a few plastic conduits nailed to the wall and run the cables through them. If you have carry power cords, they have to be on a separate conduit which is at least 6 inches away from the other conduits. This is to avoid interference. As Marsilians has suggested, be careful with power cords. Use Finolex flame resistance cables, if possible.
If you don't like the sight of the conduits, you can get a carpenter to mount a wooden panel between the TV and the cabinet that will hide all cables.
I personally prefer to have all cables open and visible so that I could fiddle with them. But that is my personal preference.
Cheers