Last October I wrote that the URL format that I had been using to read an RSS feed of various twitter accounts was being intentionally broken by twitter, but I was reassured that there was a different format (http://api.twitter.com/1/statuses/user_timeline.rss?screen_name=foo) that still worked. Now that different format is giving me errors and pointing me to https://dev.twitter.com/docs/api/1.1/overview where I read that twitter has "decided to discontinue support for XML, Atom, and RSS".
For anyone else in the same situation of wanting to read Twitter through their newsreader rather than by getting a Twitter account and rewarding their move away from standards-based networking: this time I have found the workaround myself. It is http://www.twitter-rss.com/. So far it seems to be working well for me.
ETA 6/22: twitter-rss is too unreliable. Sometimes it gives me 503 errors, sometimes it gives me ads for game apps, and sometimes it gives me lots of old copies of posts dating back months. Unless some other service turns out to work better, it may be time to give up on twitter altogether.