There have been tons of Twitter clients over the years as Twitter has risen to be the sun we all revolve around, but those numbers have dwindled significantly as Twitter has repeatedly changed its sign-in protocols and rendered non-updated clients useless. Every time Twitter makes a change, more twitter birdies fall out of the sky.
The problem is even more pronounced for Tiger and Leopard users on PowerPC where older clients that still work on those platforms are most likely to fall by the wayside. However, I found five options that still work on PPC, four on Tiger and one that requires Leopard.
For Tiger, the first client is a menubar add-on called Twit Menulet. In fact, it even runs on Panther. It's the only shareware on this list ($8) and it works as a dropdown menu from the menubar that shows your Twitter feed along with options for individual tweets like retweet, reply, and direct message--the basics. Still, it's a unique concept that just works in a way that's simple and unobtrusive.
The second, Twidget, is similar except instead of a menulet it's a widget. This is another simple client that does the basics well, although one disappointment was that I couldn't enlarge the window to show more than a couple of tweets without having to scroll down.
If you want full-fledged Tiger clients with more than the basics, though, the only options I found are twhirl and TweetDeck, both requiring the Adobe Air runtime environment. Adobe Air stopped supporting PowerPC, but fortunately you can download the older PPC version (1.5.3) here which both twhirl and TweetDeck still run on (for now, anyway).
twhirl, like all Air applications uses a lot of RAM and can be a bit sluggish to the touch. It's a bit like using a java application but it's definitely usable. In addition to the basics it comes with additional options like search, url shortening, and picture posting. It's intuitive and, though not a cocoa app, it's not altogether unattractive. Nice dock integration, too.
TweetDeck is a bit of a RAM hog as well, though it has one innovation the others don't: multiple columns. Here's where you can really use the power of a Twitter client, by browsing through groups, filtering, searching, etc. Following dozens or hundreds of people can be overwhelming, and TweetDeck helps you manage it. It's the most popular Twitter client for a reason.
There's also Syrinx, which I could not test because it requires Leopard which I don't have, but it looks to be a solid client with a nice customizable interface. There's one other client called Bosco's Reader that runs on Tiger which I didn't install because the installer wanted an administrative password, and really, I wasn't in the mood for that. Not after I read a review of a media aggregator called Elmedia that installs a kernel extension (holy crap!). Bosco's Reader is ad supported, and they seem to be aggressive about promoting it on the web. I have to say, googling "Bosco's Reader OS X" and seeing endless results with sub-headings of "Bosco's Reader makes Twitter accessible to the rest of us. I designed it for my Mom, who constantly..." was one of the weirdest experiences of my life. But I wouldn't want that to deter you from trying it.
All that being said, if you still want to experience Twitter in your web browser, there are many fine add-ons you can install in TenFourFox. You are using TenFourFox, aren't you?
UPDATE: It looks like Tweetdeck for PowerPC bit the dust. Their homepage says the latest version requires 10.6 and a 64 bit processor. Normally my sympathies don't go out to Intel users, but a 64 bit requirement for a menial application like Tweetdeck? Harsh.
In the good news department, however, I've just discovered a command line client called TTYtter. It's by the same developer as TenFourFox, and it looks really good. Has an old IRC feel to it, so I recommend you check that out.