You do not indicate whether you are a party (who was served a deposition notice) or a third-party witness (who was served a subpoena.) You also do not say why you cannot (or don't want to) appear. Your reason has to be a good reason--doctor's appointment, work, child care, etc.
The best way to get a changed date is to politely ask the lawyer for one.
If you are a party, and if the lawyer sent you the deposition notice without asking you ahead of time for a convenient day, and if you are not close to your trial date (or an important deadline), the lawyer will probably agree to move your deposition. Call the lawyer; have your calendar at hand; agree to a new date while on the phone with the lawyer. You shouldn't have a problem.
If you are a third-party witness, do the same thing, but agree to stop in at the lawyer's office to pick up the new subpoena, or to allow the lawyer to mail it to you. If you won't do that, the lawyer may not want to move your deposition because serving you a subpoena by personal service can be expensive and time consuming.
If the lawyer will not grant you a new date, then your only other legitimate choice is to make a motion to the court. But at the rate motions are being handled in LA, it will take months for a hearing, and it will put off your deposition until it is heard. If the lawyer gives you a hard time, remind him/her that when you file the motion for a protective order it allows you to suspend the deposition.
The other option--which is not allowed under the code but people do it all the time--is to just refuse to appear. If the lawyer is intractable, just tell the lawyer "no, I need a new date, I won't be there." Put it in writing and offer new dates. You could get sanctioned for your failure to appear, though, so you need to weigh that against not appearing. You want to make sure you have a very good excuse before you do that.
I am licensed only in California and this response is provided as general information only. It is not intended to be legal advice. Legal advice must be based on the exact facts of the particular situation, and by necessity this forum is not appropriate for discussion of specific, exact facts. Contact a lawyer for more specific advice. My answer to your question on AVVO does not create an attorney-client relationship.
Just pick up the phone and speak to the attorney noticing the deposition. As long as you have a reasonable explanation for wanting an extension, and as long as you agree to be cooperative regrading the re-service of subpoena (or agree that previous subpoena is valid for the new date), it is highly likley that the attorney will work with you. (If you are a party, you will not have been served with a subpoena. However, same approach should yield a similar result.)
This response is for information purpose only and does not constitute a legal advice. This response does not create an attorney-client relationship.