In California, landlords and tenants have a right to a jury trial for unlawful detainer actions. In order to preserve this right, the party requesting and/or demanding a jury trial must pay their jury fee deposit ($150) at least five days before the date initially set for trial pursuant to Code of Civil Procedure section 631(c). That means if your case is continued, the jury fee deposit is still due five days before the original trial date.
For Los Angeles County unlawful detainer actions, the Los Angeles County Superior Court has a jury fee deposit form that can be used for filing jury fees. There is no harm to requesting a jury trial and paying your jury fees because it preserves your right to a jury trial which you can later decide to waive. If you do waive your right to jury trial before the jury is called to the court room you can generally get a refund for your jury fee deposit. Failure to make a timely jury fee deposit will waive your right to a jury trial.