I agree, this can be done without a lot of trouble if you are careful, and if your original system was pretty "standard". Posts above suggest a few cautions, so I'll add one.
If you have not built a system before, you may not be aware of this important step. Every case comes with little items called "stand-offs" used to fasten the mobo into the case and create a gap between them. These things often look like short (¼") brass pieces with a threaded shaft from one end, and a threaded hole on the other. The case itself, on its rear mounting panel, has many pre-threaded holes in it. Many cases come with the stand-offs pre-installed in several of those holes, but some just include them in a plastic bag for you to install.
All mobos have mounting holes through them - very often in three rows of three holes. At each a screw goes through the mobo and into a stand-off. Look closely at your mobo in the old case and see where yours are. When you get to taking the mobo out of the old case, observe that each of these holes has little metal "fingers" like a star around the holes. When the system is assembled, the screws make contact with these fingers and ground the mobo through the stand-offs to the case itself. This is the design, but there should NOT be any contacts to Ground from the mobo at any other place (except, of course, via the power cables). That's why the stand-offs provide a gap between the case back plate and the mobo back surface.
So, ideally when the system is assembled, there ought to be a stand-off under each of the mobo mounting holes to support it but most importantly, there must NEVER be a stand-off where there is NOT a matching mobo hole. If your new case has its stand-offs pre-installed on the back plate, they will be in the most common pattern for mobo mounting holes. BUT not all mobos use that pattern. (And, of course, if your stand-offs are not pre-installed, you have to do this yourself.) So, examine your case and its stand-offs, and the mobo mounting holes, and ensure that the positions all match up. Temporarily place the mobo in the new case and compare to be SURE that each mobo hole has a stand-off under it that matches its position exactly, and there are NO stand-offs in the wrong place. If any are positioned incorrectly, you can unscrew it from the case back plate and re-position it to another of the pre-drilled holes.