If you carry the unused color around, it isn't intarsia, it is stranding. There is a fair isle technique (for stranding) that weaves the unused color on the WS to keep the floats short and prevent snags. I've done this, but it was a flat piece.
Mason, do you use the fair isle technique?
If I am correctly visualizing the intarsia in the round method that Kris posted about, then you could do it with one strand of each white and grey. If you do the black, that would need two bobbins, or maybe three.
If you use DPNs, I can see the white being all on one DPN.
I'd have to try this first.
k around in grey, stop at point to start white and join the new color.
k across the number of white stitches, then p back on WS.
Pick up grey and p back around (WS) to the far end of white, then k around RS to the white area.
That technique would work most rounds with small adjustments when grey moves one or more stitches into the white (or vise versa).
The trouble I see is you need to do something (like wraping the stitch with the other color) where the color changes to prevent a hole in the work.
Any experienced knitters care to coment on that conjecture of mine?