I've never had an issue activating the D-Pad skills with my PS3 controller. Although I don't really see a huge detriment in not being able to use certain moves while moving other than being on a frontline melee role or Bard. Even then, we are able to set up where we want our skills to be on the crossbar and we have 8 full bars to switch between, so it shouldn't really be an issue. Put the less used abilities on the d-pad and most used on the face buttons. Works perfectly for me on both WHM and Bard at least.

For example on WHM I have cures set to L2+Face buttons and attacks on R2+Face, while having important, yet less used actions on the d-pad (stoneskin, regen, repose, esuna, etc.). On bard I have the main attacks set up similarly, but replace the cures with AoE and support spells with non global cooldown actions (such as blunt arrow and blood letter). Other important actions, such as Medica II, Benediction, and songs, I set to another bar for easy access on the face buttons.

In my opinion, the original post is a non-issue. If anything, you may just need to adjust your setup and you'll be golden with a Dualshock controller.