Semantic Compositions wrote a post on translations of anime into English. In the post he said he didn't understand why many anime fans preferred subtitled anime (in which the Japanese dialog remains and is translated to English as text on the screen) to dubbed anime (in which the Japanese dialog is removed and replaced with English dialog). Being one of the anti-dub anime fans that SC mentions, I felt obligated to defend my cause and list a few reasons why I prefer subbed anime. I was going to reply in SC's comments, but after writing a few pages worth (ed.: does he ever shut up?) decided to post it here instead.
1) Dubbed anime series are more likely to be edited to change content.
Cardcaptors, the American dubbed version of Cardcaptor Sakura, was horribly mangled by the dubbers as they changed dialog, edited out scenes, completely altered the personality of certain characters, and turned a character who wasn't introduced until the 8th episode into a title character. Apparently the dubbers were trying to turn this "magical girl" show into an action-adventure show suitable for boys, but the subtitled version stays much more faithful to the original (doing no edits that I know of). Cardcaptors Uncensored is an excellent website that details the changes between the two shows, including comparisons of the episodes (episode 36 is a good example).
Yu-Gi-Oh!'s American dubbed version has also been intensely edited. Again the dubbers change dialog drastically (often adding inane blabber where there is silence in the Japanese version), completely alter plot elements (e.g. turning personal quests into cliched good-vs-evil battles), remove elements that are deemed too graphic (guns are turned into threateningly pointed fingers, scantily clad characters have more clothes painted onto them), remove any "occult" elements (e.g. five-pointed stars), change character deaths into people either just disappearing or being "sent to the shadow realm," and erase both Japanese and English text from the screen (and these were just the things my SO and I could think of quickly). The Yu-Gi-Oh! Episode guide is a good website that details some of the changes made in the dub.
2) Japanese voice acting is typically superior to the dub's English voice acting.
Japanese voice actors are generally better trained and more respected than their American counterparts. Japanese voice actors also have the benefit of working with the directors and producers of the show, while the American actors do not. Even though I can't understand much Japanese (ed.: that's an understatement), by listening to the Japanese voices I can still feel the emotions and grasp the mood that is being set. My SO and I started watching Princess Mononoke on DVD with the English dubs, because we'd heard they were good, but after about 15 minutes couldn't stand it anymore and switched to the Japanese language track, which meshed with the movie much better.
3) Dubs sometimes replace the musical score that goes along with the show, which changes the mood of the show tremendously. Yu-Gi-Oh! is a particularity good example of this change.
4) Since the original Japanese language track remains in subs, it is thus possible for viewers who understand Japanese to do their own translations. For example, I have a rudimentary understanding of Japanese honorifics (e.g. appending -chan, -san, -sama to names to indicate the social relationship between the named person and the speaker) and while watching subs I like to hear which honorifics characters are using for each other. I haven't ever heard a dub use the honorifics (which makes sense since there are no direct English equivalents, though some try to convey the same meanings), and subs rarely put the honorifics in the subtitle text.
5) Since the animation of mouth movements is designed to match the Japanese language track, dubbers must work hard to match the English script to the Japanese mouth movements (often ending up using odd constructions).
In conclusion, while subs have the limitations of any translation, dubs include many more changes to the source material than subs, and I think these changes reduce the quality of the finished product.