I wouldn't want race specific quests, in part because I'll probably only play Khajiit, Argonians and orcs, so that means there would be seven quests in the game I'd never experience, and it's not like I'm going to play a race I have no interest in just so I can do ONE quest I'd otherwise miss out on. Besides, race specific quests would be based on you're races stereotype, not what you're character is actually like. If, hypothetically speaking, the Khajiit specific quest centers around the skooma trade, for example. Well, what happens if my Khajiit has no interest in the skooma trade? I just miss out on the quest intended specifically for my chose race because it doesn't fit my character? Race shouldn't be the only thing determining characterization, so there's certainly no guarentee that just because a quest was designed you're character's race, that character will automatically want to do it.
Now, NPCs reacting differently to you based on race is fine, just no making certain quests entirely unavailable because of your choice of race. Now having certain quests only available to characters good in specific skills, or characters who did certain things in the past, or skill requirements for joining and advancing in factions, on the other hand, would be fine with me.
I am all for this and it give me more reasons to play and even try out other races.
It seems to me like if you need race specific quests to give you a reason to play different races, you really shouldn't WANT a reason to do so, because it obviously shows you didn't want to play those races to begin with, if you did, you wouldn't need an extra quest to give you a reason to choose them.