Notice of Continuance for Civil Hearing Date?

If you are the moving party, you are pretty much free to continue the hearing date on your own motion, just as you are free to take your motion off calendar.
The notice of continuance just has to indicate the old hearing date and the new hearing date, and of course, the motion that is being continued.
No, the court does not mail out any minute order because the court has not made any ruling on the continuance.

I agree with Attorney Chen. You can title the notice as "Notice of Continuance of Motion for [describe motion]." You can start the notice by stating something like the following: "To All Parties and Their Attorneys of Record: Please take notice that the hearing on Plaintiff's motion for [describe motion], previously schduled to take place on May 15, 2012 at 8:30 a.m.in Department F of the Los Angeles Superior Court, located at [indicate the address], will now take place on June 5, 2012 at 8:30 a.m.in the same department."

I agree with Attorney Chen and Attorney Lee. I would just add that the most common reasons for requesting a continuance are that you cannot be in court on that date due to a conflicting, prior engagement (if the other party is the moving party) or that something unexpected arose that would make your appearance at the hearing impossible or at least very inconvenient. If you are not the moving party, then you can ask the moving party (or his/her attorney, if he/she has one) if they will stipulate to a continuance. Most attorneys will do so as a courtesy. Even if the other party/attorney is agreeable, and certainly if he/she is not, you will need to find out what your particular jurisdiction and assigned judge requires regarding continuances. You should also check the local rules and the judge's rules (if any) as to the required procedures in that particular civil jurisdiction and courtroom.