Once you sign in using your Twitter login, TwitLAN will show you, by default, 100 Tweets and a pull-down that will let you pick which segment of your Tweets you look at (1-101, 102-202, etc.). It will load up to 200 Tweets at a time (scroll down if you don't see them right away, the screen doesn't change much). It will let you delete only the Tweets you want, but you still have to go through and click each checkbox individually.
It takes a lot longer than TwitWipe, at least you have to be involved with it longer than with TwitWipe, because you have to keep reloading the next 155 Tweets and deleting them. It's one of very few ways you can delete most of your Tweets without nuking them all, however.
TweetEraser offers more options, and is a lot easier to use than TwitLAN. It's still in beta, so it's somewhat unproven. It lets you delete as many as 3,200 Tweets at a time (though the Twitter API accepts only 350 per hour). It lets you filter messages using specific text strings, hash tags, dates or senders, in case you want to get really selective. And it is completely free, though the author does ask for donations. The donation is up to you, but if you're trying to weed out some of your older Tweets without wiping them all away, TweetEraser is probably the simplest option.
What about direct messages?
Direct Messages, which are more similar to email than Tweets, are just as hard to delete if you go through Twitter. If all you want to be rid of is some or all of your DMs (and replies from your correspondent), there are more options.
DMWhacker is the most widely known (and, at this writing, busiest) of the applications listed here; once you get in the tool itself is pretty fast and gives you the option to delete DMs by sender or date.
TweetWax looks approximately as fast as DMWhacker but lets you sort messages by tag as well as sender/receiver;
InBoxCleaner has fewer options, and only lets you delete 10 messages per day for free. More than that are $19.99.
Twitter, for its part admits it doesn't provide a way to delete more than one message at a time. Its best suggestion for cleaning out some or all the Tweets in an account is to create another, temporary account, move the Tweets you want to save to it, delete the old account, then rename the new account with the old name.