The first, and most official option is simply to rent foreign movies.  Get yourself an account at Tsutaya or one of the smaller chains (you'll need some form of Japanese ID for this), and you can rent movies to your heart's content.  You can find Chinese, European, English, and Korean DVDs at the rental shops, and on almost all of them you can choose the audio and subtitle languages.  This is also a great way to build your Japanese skill.  The flagship Tsutaya shop in Shibuya is unrivaled in both size and popularity.  The drawback is that movies aren't released until after they are translated, which is usually significantly after their original release overseas.  Tsutaya also has a rent-by-mail service called Discas similar to NetFlix in the US, which is convenient, but you'll have to know enough Japanese to navigate their web page and enter the movie titles in katakana.  

There are of course movie theaters, and they play foreign films as well.  The audio usually isn't dubbed, with subtitles preferred, but this varies by movie and theater.  Note that nearly all movie theaters charge the same (high) price of around $18 US, so you may as well go to the fanciest one you can find.

Through the magic of the internet, you watch whatever you want from the comfort of your own home and much closer to the original release date.  Tsutaya has an online service, but it has the same delay problems as renting DVDs, requires Windows (Sorry Mac and Linux fans) or a special TV, and is not for the faint of heart.

Obviously there are the less-than-legal P2P applications, newsgroups, and BitTorrent, if you are in to that.  Note that some ISPs may cut you off if they detect this sort of behavior.

iTunes is actually a great option.  From the Japanese iTunes store, your selections for TV and movies are extremely limited, but from the US store, there is plenty of content to go around.  Technically, you can only use the US iTunes store from a US account, which you can only get if you live in the US.  The good thing is that they validate this based on the billing address of your credit card.  If you have a US credit card (or check card), you're golden.  If not, you can always buy US iTunes store prepaid cards online.  You can even get an AppleTV to watch iTunes content on your TV if you don't want to hook a computer up to it.

Web Sites: There are web sites like BBC, FanCast, Hulu, and Joost which will let you watch European and US content online, however some of them attempt to restrict viewing to people they believe are in the US or Europe based on IP address.  This is for legal reasons (i.e. they may have distribution rights only in the USA), but is unforgivable on many levels.  The contracts for such things should be updated from the dark ages to the internet age.  In fact, I sometimes got the "We're sorry" message when I tried to access such sites inside the US!  At any rate, you can access such pages by a variety of technical methods, including routing your internet access through a foreign VPN with servers located in Europe or the US.  If they manage to eventually completely block that (not likely), then I am sure people will feel all that more justified about downloading shows illegally.

Note that the region code on Japanese DVDs is different from the codes in the US and Europe.  This means that unless you brought a DVD player with you, you can't play them.  Computer drives can be switched to play foreign DVDs, but then they won't play disks made for the Japanese market anymore.  You can only switch the region code of a DVD drive so many times, and then you will be stuck with whatever it was set to on your last change.  Some DVD drives can be hacked to allow unlimited changes with special software, however.