I'll answer both parts of this question and especially your latest question.
Thank you notes - if you have gone through an interview, our advice to our clients is to always submit a written thank you note to interviewers within the hour. So write and drop it immediately after the interview. That way they know you did it immediately (post mark date) and you get the added benefit of being in the acceptable window of time. A handwritten note is a great way to add that little personal touch. We don't recommend fancy stationary, but we do recommend good quality.
(On a side note, if your hand writing stinks, have someone help!)
As for your following up question - I think a quick email is perfectly acceptable. "Last time we spoke you had mentioned I should look for correspondence within a week and I just wanted to follow up". Really not a big deal. You won't come across as desperate. Desperate would be 1 week + 1 day. A week or so after the date they promised has passed is just you being responsible.
I hope that helps!
Bhavik