Thank you all for your responses, however, maybe you missed the main point of my question (or I didn't state it clearly enough). That's good to know I can set up rules to move messages to IMAP folders, however, this requires keeping Mail running at all times in order to sort them and apply these rules. If Mail doesn't run, messages don't get sorted, which is the crux of my question. What if I don't want or can't keep Mail running 24/7 on any computer?
Hypothetically let's say I do keep Mail running on my main machine 24/7, and let's say I set it to check for new mail every 30 minutes (and therefore, every 30 minutes it applies my rules).
Let's say I'm away from my main machine, and from my iPhone I check my IMAP mail 10 minutes into the 30 minute cycle. That means on the iPhone I'm going to see all my unsorted mail -- which is a disappointment...and that it will take 30 minutes before my main machine's Mail program will sort them. (But I don't want to set my main machine to check once a minute because that will just annoy me and prevent me from getting work done when I'm actually working on my main machine.)
Should I set up the same rules on every computer I use? What a pain to have to do things in triplicate, and then anytime I want to make changes, have to do them in triplicate too.
I realize the question I'm asking goes beyond Apple Mail, but I'm asking if there are any other ways to handle stuff like this, as I make this transition from POP to IMAP, and from one main computer to three.
(BTW my mail accounts are not Mobile Me accounts, before anyone asks. They are hosted by a web site hosting company.)