Gotta say I'm quite surprised I haven't gotten into the habit of doing that yet. The only time I've skipped through a mission briefing is if I'm in a group doing a Heroic quest and don't want to hold everybody up while I grab this random mission giver that I passed on the way to the Heroic. Otherwise I've been very interested in following the storylines of the various planets.
Also, there's a few characters I've met that have had memorable interactions with my character and I'd have never had a chance to experience that if I'd just skipped through all the conversations. Like the Sith on Taris that I wound up hating so much that I took a few Dark Side choices to get back at her during my questing there.
I wish she'd have dropped a lightsaber or something I could keep as a trophy instead of some lame green gloves...
