Product activation (most notably on MS Office from 2002 on, and Windows from XP on) will often notice that the the mobo has been changed and require that you reactivate. Normally if some time (several months) has passed since the last time it was activated, then it will often reactivate over the internet without any further problem.
Sometimes however it might refuse and you will need to phone MS. Sometimes the phone based automated activation will then pass it. Other times you may have to go as far as a real person.
What happens then depends on what you say, and what you are activating. In the case of an OEM version of windows (i.e. not one bought at retail) you are in effect in violation of the terms if you are simply "upgrading" a machine - since that in effect is a "new" machine, and OEM copies are supposed to die with the hardware they were shipped on. The exception to this rule is if you are repairing a machine under warranty. So explaining that you have just repaired a customers machine with a new motherboard etc, will normally get them to reactivate.
With retail versions you normally just need to assure them that you only have less than the licensed number of installed copies already, and they will reactivate.
(Note with Office 2013 when it was first released they quietly dropped the retail license terms - forcing OEM terms on all copies. However after a bit of an outcry they backed away from this and the PKC versions (i.e. medialess boxes just with a key card) can be reactivated under retail style licensing now.
In the case of many machines you can simply use the number on the CoA sticker. If you need to recover keys then there are various utils out there like magical jelly bean and produ key that will recover the original key from the installation.