Not a dark brotherhood kind of guy I take it?
Nazeem may be annoying, but he's harmless. I wouldn't kill him unless I was doing a psychopathic Dark Brotherhood character, or a playthrough that was completely for [censored] and giggles.
Nazeem's easy to dodge, and Amren is actually one of the few NPC's to change up his lines once you've done his quest (shocking, I know). The one who's always screaming his lungs off about being the chosen one generally goes into early... retirement.
Life is not always what you'd want it to be and you cannot kill people just because they are annoying, even if some deranged nuts actually do it in real life. I suppose a video game like this one is a good way to learn patience.
Edit: oh, by the way, if you are on pc, you can silence everyone very easily by putting one game setting to 0, "fAIMinGreetingDistance".