Connect neutrals together in 3-gang box (load side) with 3x 3-way switches?
Neutral is not ground

Neutral is the normal path of current return.  Ground is an emergency path,  only used if something is seriously wrong (in lieu of sending power through a human, and it helps to get breaker to trip).  Because of that, it's OK to spiderweb all grounds to each other.  This should never be done with neutral. 

Currents must be equal in each cable or conduit

Because AC current in wires causes AC magnetism, and the wires' magnetism must cancel each other out.  If it doesn't, it will cause eddy current heating, and noise/vibration leading to fatigue, breakage and arcing, both of which can start fires. 

It's a lot to sort out. Mark the wires.

They sell 5-packs of colored electrical tape for about $5 at the hardware store.  Typically blue red yellow green white. 

So you might do something like this in box 1



and this in box 2



*The neutrals will be a wire from each cable that need be wire-nutted; I simply didn't bother drawing the wire nuts. 

Note how the 2 travelers are color coded the same.  That is because there is no need to distinguish travelers from each other.  

I also spiraled the travelers' markings quite some distance, but the switched-hot only the width of the tape. 

Now that looks super clear on the drawing, however in the box, things may be so confusing you need to mark the neutrals too. But you cannot use red, yellow or blue (hot colors) on a neutral wire, or you will inadvertently redefine it as a "hot" wire, which is wrong.  White, gray and green don't have that problem.  They won't redefine a hot because you're not allowed to redefine a hot to a neutral or ground.  White and gray are neutral colors, and green won't redefine a neutral to ground, because that's not allowed either.  So brace for an argument with the inspector, but I say using white, gray or green on both hot and neutral does not change the purpose of the wire, therefore is only marking for identification.