I find Java's Date API usable, to be honest. Most of the issues I've seen and heard about relate to the verbosity, the need to involve multiple classes to do anything useful (Calendar, Date, DateFormat/SimpleDateFormat) and the lack of simple accessors like getDayOfWeek().
Joda Time is a well-respected alternative API in Java, and in the Why Joda Time section it gives some more arguments as to why it is a viable alternative that might be of interest.
