Is safety really as big of a concern as people make it out to be?

Hey all. I'm visiting Seattle soon for a few days. My plan, besides some of the more outwardly touristy stuff and touring the University was to just walk around a bit, find some nice parks, ride the light rail, and maybe explore some neighbourhoods. However, I can't help but notice all of the things I hear about crime and safety, both in downtown and some other neighbourhoods. Is it really as bad as some say it is? Would it really impede my plan at all?