Well, a CPU box should be more visible on an x-ray then a laptop since there is a lot of open space inside, so chances are good that the screeners will be able to clearly see that it is not packed with explosives (at worst, they will want to do a swab test). I have not had to power-on a laptop at a security station since about a year after the Lockerbie Bombing. At most, they swab my PowerBook (but never my Wintel units).

If it meets the dimensions and fits in the overhead, you should be fine. And your LCD in a bag would meet the "one personal item" rule, so should not be an issue there, either.