The first things I would check are:
1) Are the converter box and TV grounded properly? If there is anything wrong the with grounding of either device or with the ground wiring of the house, there can be a "ground loop" situation (http://en.wikipedia.org/wiki/Ground_...electricity%29
2) Could be noise on the power line. Can you test with a power strip that has EMI filtering?
3) Go through a process of elimination to try and isolate the cause. For example, swap out the converter box and feed the TV with a different video source (like a VCR, DVD player, or a different converter box) that has the same kind of connection. Try a different set of cables. Take your converter box to a different TV and try it.
A few questions:
Do you have any amps or any other powered devices between the converter box and the antenna?
Is the connection between the converter box and the TV using "RF out" or "video out"?
This does not sound like an antenna issue or a problem with signal detection. If you are getting a picture out of the converter box, it means that it is detecting a clean enough signal (good SNR) to decode the digital data and play the video streams. If there was a problem locking on to the signal, then you might see picture drop-outs, freezing, or some blocking artifacts, but would not see diagonal bands like the ones you're describing.
This seems to imply that the problem must be in the converter box, the TV, or somewhere in between.
Electrical problems can cause symptoms like the ones you're describing, so that's why I'd check those first.
You didn't mention having an amp, but if you do, you want to make sure that you're not somehow sending power to places it's not supposed to go (some amps are powered via the same coax that carries the signal, so you want to make sure the power is not being sent into the converter box or any other device that isn't expecting to receive power that way).
It could also be a bad converter box, bad cable, bad connector, or maybe the TV is going bad.
Although "interference" from other signals (FM, AM, TV, etc.) might be possible, the chances of that being the case seem to be exceedingly remote. You don't happen to live right in front of a transmitter facility, do you?