If the question is whether or not you need to disclose your former name if asked, the short answer (unless you changed your name at a young enough age that you wouldn't have any background-check-relevant records under the old name) is (unfortunately) yes. If they want to verify your work or education history, and they don't know the name the institution has you down as, you may get a (false) response that you never worked or attended school there (and assume you're making up that history) - likewise if they try and contact a reference that doesn't know you by your new name. It's even more critical if they check your criminal or (to a lesser extent) credit history - if you end up (even unintentionally) concealing an offense or unfavorable credit entry (no matter how minor) because you didn't provide the name/alias you committed the crime or obtained the credit under, that becomes a material omission with the possible consequences thereof. Employers are not supposed to ask that question merely for the sake of knowing your former name, but if they need it as a practical matter to properly check your background then you can be required to provide it.
If the checks are being done by a third party separate from the one deciding on your hiring, you could try and see if you could provide the sensitive information straight to them instead of on the form that goes through the employer (this might be helpful for jobs that require a "formal" criminal history check where another agency pulls your history).
