The POSIX equivalent for what you described is POSIX condition variables. Note that condition variable must always be used in pair with a POSIX mutex, but quite frequently several condition variables use the same mutex, so if you aren’t going to use the mutex exclusively for the condition variable, you shouldn’t place it in the class. The mappings by meaning in your case between Win32 and POSIX API should be:
CreateEvent -> pthread_cond_init
CloseHandle -> pthread_cond_destroy
WaitForSingleObject -> pthread_cond_wait or pthread_cond_timedwait
SetEvent -> pthread_cond_signal or pthread_cond_broadcast
Fortunately, there is a lot of documentation regarding this, though I recommend the fundamental Programming POSIX Threads.