If you have one then it would not hurt to use it. If not I would not be too worried, just take sensible precautions such as grounding your self and the cases before you start, and not wearing any cloths that generate too much static.
With you old case with the PSU installed and connected to the socket (It doesn't have to be switched on) touch some exposed metal, touch the new case against the old case as well to ground that. Failing that just touch a piece of exposed copper pipe leading to a radiator / tap.
Unless you wear 1990's shell suits static should not be a problem. I have moved a few motherboards into new cases and have never used a wristband and so far have been OK. I always worry about knocking the capacitors more than anything.
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