If the new case supports they form factor of your motherboard (ATX m-ATX, mini-ITX), then you can easily take parts from one case and stick it in the new one. Also make sure that the new case supports the power supply you have. If it won't fit in the new case, you might need to upgrade the PSU.
One thing to look out for in a new case if the front panels headers, such as USB 2.0/USB 3.0. If your old motherboard doesn't support them: Like if the old board has USB 2.0 headers, but case only has USB 3.0 headers, you will need to get an adapter for this. Sometimes the case manufacturer can provide adapters to work with old motherboards.