I would definitely contact them. I don't think it comes across as desperate just keen and organised. Sometimes in a big company there is an assumption that someone else has done the follow through on interviews and it does not always happen in as timely a manner as it should!
I've had two interviews for a job recently. Both went really well, as did a series of job related tests. They said they wanted to fill the position quickly and would contact me soon. It's now two weeks since the last interview and I've heard nothing. I'm guessing they may have selected another candidate, but I'm wondering if it would be bad form to contact the company's HR department and ask where I stand. I'm very keen but don't want to appear desperate!