If the date they promised has passed, follow up once, politely, referencing that date. Ask for an updated timeline rather than a decision — it is easier to answer, so it gets answered.
Within 24 hours, while the interviewer still remembers the specifics of your conversation.
Email, almost always. It arrives while the decision is live. A printed letter is a nice supplement for traditional industries, never a replacement.
Send each of them a separate note referencing what they personally asked about. Never send the same text to everyone on a panel.