I never really watched Cartoon Network but I found out that it shows several Japanese animes, so I have started to try to catch some on TV.  So far I have seen only a few episodes of Inuyasha and I have yet to learn about each characters and the storyline, as it's difficult to watch the episodes in sequence.  I just wish that the shows aren't dubbed, as I don't like how the English translation sounds.  I'd rather hear the original sounds by the Japanese cast and read the English subtitles.

Adultswim shows that I am interested in watching (text from Cartoon Network link):

COWBOY BEBOP (TV-14)
Taking place in a future where Earth has migrated into space, "Cowboy Bebop" follows the exploits of bounty hunters, Spike and Jet, as they roam the galaxy tracking down criminals in a world rampant with crime and opportunities.

GHOST IN THE SHELL: STAND ALONE COMPLEX TV-14
In a future when cybernetic enhancements are a way of life, Major Motoko Kusanagi and Section 9 take care of the jobs the police and the army can't handle. Hackers and sharpshooters, detectives and cyborgs, Section 9 is the one thing that stands between cyber-criminals and their prey. Based on the classic anime movie Ghost in the Shell, Stand Alone Complex is a combination of stand-alone episodes and episodes that are just a part of a bigger, more complex story.

INUYASHA (TV-PG)
Kagome Higurashi is on her way to school when she is abruptly transported through an ancient well into feudal Japan. Surrounded by monsters and magic, Kagome teams up with the half-demon InuYasha to recover the missing pieces of a magical gem that could cause disaster if it falls into the wrong hands.

LUPIN THE 3RD (TV-14)
Lupin the 3rd is suave. He's debonair. He's also the world's most notorious thief. Traveling the globe in search of his next big score, Lupin does not hesitate to get himself into dangerous capers, even with lovely ladies and Interpol agents hot on his heels.

REIGN: THE CONQUEROR (TV-14)
In the ancient world of Macedonia, science and magic are one. And a single man controls them both. Reign: The Conqueror is a science fiction retelling of the adventures of Alexander the Great, featuring character and concept designs by Peter Chung, the creator of Aeon Flux.

TRIGUN (TV-14)
Vash the Stampede is the most notorious gunslinger on the futuristic desert planet of Gunsmoke. He's got a giant price on his head, and there are lots of disreputable types out for the reward. Little do they know, Vash is actually quite a good-natured sort and a bit of a doofus. Just don't get in his crosshairs

Non-Adult Swim programs:

Astro Boy

Cyborg 009

Dragon Ball

Dragon Ball Z

Dragon Ball GT

Duel Masters

.Hack/Sign

Hamtaro

Knights of the Zodiac

Rave Master

Rurouni Kenshin

Yu-Gi-Oh!

Yu Yu Hakusho

Click here for the link to Cartoon Network's show schedule.