GFI Breaker on a three line 240V circuit
How it works

When you have multiple poles or phases and you need to GFCI-protect, the only possible way is a single GFCI device which has all current paths running through it.  

Current should confine itself to the three wires.  An axiom of AC mains distribution is that currents must be equal in each cable or conduit, i.e. current out must equal current coming back (so EMFs cancel each other out, and you avoid vibration and eddy current heating).  A GFCI is simply a sensitive detector of that: all conductors go through the same current transformer.  

In split-phase, you might have L1, N and L2 currents of 15A, 2A and 13A, respectively.  That balances.  

If the currents don't cancel out, then current is using a third -- er, fourth path in your case. The GFCI doesn't know what the fourth path is, but in case it's a human, the GFCI trips. 

What to do

So yes, mechanically you need all 3 conductors to go through the same GFCI device.  Nearly the only option on the table is a 2-pole GFCI breaker.  The three wires go on the  three breaker terminals, and if it's not plug-on-neutral, the white curly pigtail goes to the neutral bar.  (the GFCI pigtail is needed even on 240V-only circuits, to power the GFCI itself; once a Filipino imported some US 2-pole GFCIs and had to create a 120V source inside the panel.) 

Remember, breakers must be a match for the panel. GE->GE.  BR->BR.  Murray->Murray.  HOM->HOM, unless they are UL cross-listed (BR->Challenger C) or classified (QD->QO or CL->T&amp;B) breakers.  Don't stick a HOM breaker in a BR panel; the buses don't properly fit and will arc. 

With a 30A breaker the cable must be 10 AWG or larger.  The equipment's instructions must be followed re: minimum wire size and exact breaker size.