I think I'd go for the same white/black for all. And would look for a tone on tone for each.
Though that would depend on what greens you have chosen as to whether it'd look better with solids or the tone on tones. Also, dependent on your greens a cream/ivory may blend in nicer than the white, though a bright clean white would likely help your fabrics pop out more. Depends on the look you want too.
YOu'll have enough going on with all your different greens, and that's what you want to be "seen" in the end, not the background. So go for consistency for your black/white.