I asked this question of a younger programmer the other day, because it seemed to me, that to HIM, learning C was a rite of passage, and that he was less of a man for not knowing it.
I find macho philosophies in software development both amusing and counter-productive.(If you really want to be macho, become a lisp hacker). My amusement may be personal, but the counter-productivity of using C, when better abstractions (i.e. programming languages) already exist, is real. It creates fiefdoms and priesthoods that are counter-evolutionary and hard to maintain, and leads the death of much software.
Personally, I would rather std::string be pored over by many eyeballs and evolved than change strcpy to _strcpy or strlcpy. Unless I am working for NASA on an embedded device for a satellite, I would rather Moore's law or SMP or DSMP give me the speed I need, than give up productivity to squeeze every last CPU cycle. Developer time is a lot more expensive than hardware (except on satellites and space stations).
Apologies if it is your ambition to work for NASA on embedded devices in space stations, I just think you may as well learn C++. You get most of C, plus some really useful and productive abstractions as well.
If you want speed, learn inter-process communication and the principles of symmetric multi processing. With C++, you also get to abstract away the problems of strcpy and strlen, replace Byzantine function references with class methods, and 35 parameter functions with polymorphism. Best of all, most of the programming world will still think you are manly if you know C++, so you get that too.