Do you track time at your job? If so, how? If not, why not?

Previously at my job we didn't track task time, but that has changed. We don't bill hourly, but per project & we have a project management system. However, now we are required to track time per task, our internal morale has fallen through the floor. Is it a good idea to track time if we aren't billing hourly, or is there a method to track time without destroying internal morale?