Short answer? No. A new Motherboard wouldn't fix the issue.
The microphone is actually soldered onto the Dock Connector, not the Motherboard. As such replacing the Motherboard wouldn't do anything to address the issue as the broken microphone will remain.
The removal and installation guide for the Dock Connector is located here:
iFixit even sells the replacement part:
So if you feel comfortable replacing the Dock Connector yourself, that is your best bet.
It isn't a particularly easy task, but it's certainly doable.