It's actually kind of convoluted how it works. There's a difference between crimes (wrongs against the state, e.g. larceny, murder, kidnapping) and torts (civil wrongs, i.e. the kinds that you can sue for, e.g. negligence, economic fraud, defamation). Some actions (such as assault, battery,etc.) can be both though (since the state can prosecute someone for assault, and the person assaulted can sue the assaulter as well for compensation).
People who are tried for crimes are tried in the state(s) against which they committed the crime (if they violated a state statute), or in federal court (if it was a federal statute), or in both in certain cases.
For torts (in general defined as committed when:
1. a duty imposed by the law owed by one "person" (entity) to another
2. is breached and
3. proximately causes
4. injury or damage to the owner of a legally protected interest
which is what the RIAA is suing people for), on the other hand, the plaintiff must show that the courts they are suing in has 1) jurisdiction over the parties, and 2) subject matter jurisdiction over the case itself.
Jurisdiction over the parties BASICALLY means that both parties have to have some "CONNECTION" to the state in which the court resides -- there can be "no surprise" to either party if they are called out to that place. Courts automatically have personal jurisdiction over the plaintiff because he/she is the one who brought the suit to that court -- therefore there's no way he/she could be "surprised" that it is being tried there. For the defendant, the "minimum contacts" include things like presence (whether he/she has been there/whether the suit is served while they are there), residency (whether he/she lives there -- or if it's a company, the state in which it's incorporated), consent (whether the defendent "agrees" to be tried in that state), or OTHER minimum contacts (basically otherwise "sufficient" affiliation with the state in question to make it reasonable that he/she is tried there involving things like property and commercial activity). [Edit: I found some more info on minimum contacts specifically w/ regards to internet cases:
http://en.wikipedia.org/wiki/Personal_jurisdiction_in_internet_cases ]
So basically, yes the RIAA can do what Jon`C mentioned (there are more details involved than the three-days-you-can't-show-up-so-you-lose-your-trial thing though), because their lawyers can make the argument that you have a "connection" to wherever it is they're suing you at (omg you downloaded a song through a server that resides there?! or something like that. I'm just generalizing/speculating on this though... I haven't actually looked into these cases.)
Also note that if it is blatantly clear that a court does not have personal jurisdiction over you, you can send a lawyer to dismiss on grounds of lack of jurisdiction.
** Disclaimer: I'm drawing on some pretty basic legal studies knowledge and I am not the best person to provide entirely precise or accurate info on the matter... it'd probably be better for someone who's extensively studied the relevant tort/jurisdiction laws to speak on it further. Don't take this as legal advice because I will probably screw you over**