The textbook answer is 'it will kill components' which is technically true.
The anti-static wrist-strap thing is a bit of a myth though, if your PSU is plugged in, then touching any part of the case will ground you instantly, as will touching a nearby radiator or what ever. I have blatantly mis handled many parts which are all fine.
Bare chips, or mis-handeling components, in particular, RAM or PCB's with RAM chips on them, like graphics cards, can be a bit dodgy, and as such I awlays make an effort to ground myself at regular intervals if im handeling something expensive, but that said...
...I was striping down my media PC for a spring clean, which is an s939 A64 rig, and being used to intel rigs, I forgot that the chip was locked onto the socket, (I thought it was some TIM that was sticking it to the socket) so it levered it out with a screwdriver.
The chip flew across the room, and suffered some bent pins which I fixed using the credit-card method, and she still runs fine to this day.
So go figure
But better safe than sorry and all that. Just be carefull.