I'm pretty unhappy to hear that Bloglines is shutting its doors. I really liked this service, and I used it quite a lot over the past few years.
I find the rationale for this site's closing to be...strange. I can't argue with Ask.com's claim that traffic to this site is down significantly. However, I do find the explanation for this trend to be weird: apparently I am supposed to understand that "everything is social now" and "RSS is dead". But, I have a hard time accepting this.
I use RSS technology exactly for its designed purpose: to make it easier for me to keep track of a lot of web-content, without me having to actually go out and poll each site for that content. For this purpose, RSS is wonderful.
I'm pretty dubious that I'll ever be able to keep track of the content that I am interested in by somehow interacting with a social network. It's not that I'm not social -- it's just that I have my own set of interests, which I assume are fairly specific to me. I don't think that I ever will find myself in a social network in which I can interact with my "network" and thereby keep track of all of the interesting content that I like to read.
So, now I need to find a new RSS reader. I haven't found one yet that I really like. Bloglines had a pretty reliable site, and this site gave me a nice hierarchical view of the feeds that I subscribed to. I haven't found anything yet that works as well as Bloglines always did.