It's probably interlacing. If you don't understand the difference between progressive and interlacing, do some googling. It's very basic video stuff. Select a clip in the browser or timeline and hit command-9 and tell us what it says for field dominance. If it's anything but "none" it means that the source is interlaced and computer displays are progressive. When viewed properly via a video card on a video (not computer) monitor, it should look ok. If your end use is for computer display, you should deinterlace your material either before editing or after (your final output). Compressor will do a much better job of this than fcp. Post back if you want some help with compressor.
Also, the camera records a highly compresed version of video. While it looks like FCP will accept these files in their native format, it really does not like them nor does it edit them without issues.
Best practice is to convert them to ProRes LT before bringing them into FCP. If the camera is supported by FCP, use Log and Transfer. If it isn't then Compressor or ClipWrap will usually work.
Once the material is in ProRes, you'll find editing moves much more smoothly with few issues on output.
You say above: "When viewed properly via a video card on a video (not computer) monitor, it should look ok." I'm not sure what you mean by a "video not computer monitor". I edit on an IMac which had its video card and screen. Wondering in what way this computer's screen is not a video monitor.
Two years after the post above, I see, but thought I'd reach out with this question anyway.