My wife is a writer and often has pieces published on other sites. The articles usually list a short bio and her blog URL but never seem to anchor link to her blog. They will write www.blogtitle.com in the bio but only as plain text. My question is does this count as a backlink to Google for SEO purposes? I'm guessing it doesn't so I've been trying to convince her to request a proper link back to her blog.
No, according to Matt Cutts (the Google alter ego) the url text with no link it does not count as link.
I think the undesirable fashion of using text url instead of links might be caused by Google nofollow policy. Basically those articles once would have probably added a nofollow link to your wife's blog because nofollow links apparently did not leak out link juice. Once Matt Cutts explained that nofollow links do still leak out juice, many eager webmasters got scared to loose a tiny, tiny bit of juice and started to use text url instead of adding links.
Does it count as a backlink? No. But not all is lost. These are citations and count nearly as much as a link some say. I am skeptical, however, I have seen citations work positively. In fact, any link created on another site may end up being no-follow anyway which is treated differently. Please do understand that no-follow does not mean no-follow (from a spider's perspective) if the target site is known to Google or becomes known to Google via other means and that the target page is not restricted in the robots.txt file. No-follow is meant to be a protocol to prevent blog, forum, and comment sites from becoming targets of abuse.
The upshot? While it is not necessary to create links in the case that you described, it does help and it will count more. There is no significant harm if the old citations are not updated to links. But if you can edit them, I would. Should you create links from this point forward? Yes. Absolutely.