Can I split a 208/120V 3-phase circuit into 3 separate 120V circuits and keep the shared neutral?
This only works with genuine 3-phase power.  If you are using a "phase converter" to fabricate "3-phase" from a single-phase supply, Do Not Do This. 

What you would be creating is called a multi-wire branch circuit (MWBC).  There are several articles on here on how to do MWBC's, such as mine here, or you can Google it.   You will need to pigtail neutral, mark the wires as grouped wherever they are near other wires, and as Kris points out, use a 3-pole breaker. 

The circuit breaker would need to be sized for the thinnest wire including the ground.   Sometimes 4-conductor circuits are installed with a smaller neutral than the hots, in that case the smallest wire decides the breaker size.  

Also if you are feeding common USA 120V receptacles (i.e. NEMA 5-15 or 5-20), the breaker can be no larger than 20A. There are special rules which permit 30A for lighting circuits.