I want to state for the record that I am meticulous about keeping my blog updated and thus hack-free, I do not use it for any kind of nefarious SEO purposes, and I don't even run advertising. It's just my blog.
In that case, would 301 redirects help or hurt? Am I better off just starting fresh, and hoping that anybody who cares enough about my content just updates their links?
It must depend in part on how people use your site and what you hope to happen with the move. If most of your users just go to the top front page every day to read what you've posted, you could almost certainly post the new URL on your front page (maybe not even link it and make them copy-and-paste it), a one-time slight inconvenience for your users and you start from zero on accumulating new Google juice. After a little while, you'll start showing up in Google rankings again.
If a lot of people have bookmarks and/or on-page links to your site, and you have a bunch of regular users that you want to transition, a 301 is the web-standard right way to do that; but the cost is that you'll have a longer walk in the Google wilderness. I have to assume that the "transferred from a bad neighbourhood" poison isn't permanent (especially if you take down the 301s after a few weeks or a month) but it seems like it would start you off with negative Google juice.
Which might be worth it, or might not. How many referrals were you getting from Google before the co.cc thing went down?
In that case, would 301 redirects help or hurt? Am I better off just starting fresh, and hoping that anybody who cares enough about my content just updates their links?