All the Twitter accounts I'd ever want to follow have fairly infrequent status updates so I just add them to my RSS reader and follow them that way instead of having to interact with Twitter's website or application in any way.
that's what I used to do, when google reader was still around.
Twitter also removed direct RSS support for tweet feeds, and I did't like the hassle of having a third party in between.
It'd be really awesome if there was a tweet-ish open standard that anyone could implement, ala email or RSS (but supporting encrypted/password protected feeds). I feel like Google would be the perfect company to do it, but they decided they don't want to pander the nerds, and instead force G+ down our throats. (G+ I like. I just don't like how no one else is there)
