First thing I did was cleared the cookies, but I think the problem wasn't so much that.

Using the link in the article that was posted by Ryan seems to be what we most likely all have in common.   

Instead, goto http://www.battlefield-heroes.com and log in using your EA information.  Then clicking on play now -should- start the download.  

Make sure to scroll down to the end of the user agreements before you click OK. I'm not sure if this is necessary, but it doesn't hurt to be sure.  

I'm pretty sure the most important part is making sure you are logged into your EA Account.  If I can help I will, but I'm computer-stupid.  But when I did all of the above, I got it working for me.  Hope it helps.