I have played with zounds of races on Morrowind and a bit less on Oblivion. Never played Khajiit and Argonian, so went with Khajiit for a stealthy run this time (+ the Thieves Guild is always my favorite guild

). Didn't start over once so far, am about 25 hours into the game.
In Oblivion playing Redguard warrior was perfect for me, still think that suits me best (and training lockpicking to get in places I shouldn't be getting into - which is nice). Played typical Nord in Morrowind mainly using Keening and Sunder while being pretty much invincible at level 90 (big scary bold one, looked like Nappa from DBZ if you remember him

), also played Dunmer, Bosmer, Altmer, Imperial and tried once with Argonian on Morrowind, but the last one didn't last long.

Considering the races didn't change that much over the course of the games (ability-wise), I didn't need to think for too long when I was looking for 'something new'.