Don't laugh at me.

In the original sentence "Don't laugh at me.", the speaker is expressing a desire not to be laughed at. This sentiment is correctly transformed into passive voice as:

Option D: "Let me not be laughed at." - This correctly reflects the passive construction where "me" is the subject and "not be laughed at" is the passive verb phrase indicating the action to be avoided.

Option A: "Let me be laughed at." - This option incorrectly changes the meaning, suggesting the opposite of what the original sentence intends.

Option B: "Let be me not laughed at." - This is grammatically incorrect and does not form a valid passive voice sentence.

Option C: "I am laughed at." - This changes the meaning and structure of the sentence, making it an active voice construction.

Therefore, the correct answer, maintaining the original meaning, is Option D: Let me not be laughed at.