In accordance with magazine Wired, this is the strategy that Lawrence Lessig recommends to use in desperate situations:
1. Compile the email addresses of all those persons to which you have not answered yet. Then copy all of them in the field Cco of a new message that you will address to you.
2. Write a respectful note to explain them the squeeze in which you meet. Apologize e.g. write five times mea culpa in equal number of paragraphs - and promise that in the near future you have the commitment to be punctual answering your e-mail. Try to be as credible as you can.
3. Ask them if there is any urgent matter to forward it to you ASAP and promise to them that you will focus all your attention in it..
It seems to be a good strategy.