I have no particularly strong thoughts on this, but the Chronicle published a controversial piece last week titled "Bloggers Need Not Apply"; its author describes a job search in which various candidates were ruled out because of what they'd revealed online in their blogs (or perhaps for the mere fact of their blogging? It's not entirely clear). Some good discussions of this on various academic blogs have ensued; excellent posts here and here, and the first link came from Steve, who also has sensible discussion.
I think it should be possible to blog under your own name as a graduate student or untenured faculty member (well, obviously I haven't chosen to remain anonymous here). Anonymity is difficult to maintain, and I think may sometimes offer a mistaken impression of safety or privacy that later may explode if the blogger's identity is revealed. I dislike the Chronicle's habit of having many of its columnists write under pseudonyms, and I find the pieces by writers like Dennis Baron and Stanley Fish that are published under their own names (with details changed if there are issues of confidentiality etc.) the most valuable. My rule here is that I don't write anything that I wouldn't be willing to say at, oh, a party where 5-6 people were listening to me and others present could potentially also overhear what I was saying. But then I am also not writing an academic blog, though I am an academic; concerns about the privacy of students and colleagues mean that I rarely even allude to anything going on in my work life. The privacy obligation is also asymmetrical, by which I mean to say that while I might not like to see a long post by a grad student complaining about a meeting they'd just had with me, it would certainly be within the student's rights to post about it, whereas it would be wholly inappropriate (I'd say even unethical, whether or not I named the student) for me to write a similar post about a student. On a slightly different note, search committee members shouldn't blog about candidates, but while candidates may technically be within their rights to post about job interviews, they should also remember that if they blog under their own names, it is entirely possible that a search committee member will see that discussion afterwards and that it will strike that committee member as unprofessional or off-putting. Just a few thoughts, anyway. I'd be interested in hearing from others in the comments.