Well, it's about time that I had a voluminous go at this. Many answers here are great, but I feel that the issue is very simple. Riddles are word puzzles of the greatest sort. They simply have one drawback. Anyone can make a horrible riddle; it's stupid easy. Math or science related puzzles as an example require that the puzzlesmith be reasonably competent in a field, whereas a riddler need only type a bit and think himself clever. If riddles are allowed on the site, they are likely to always be the greatest source of crap needing cleanup. I don't dispute this.
Here's why getting rid of riddles is a terrible idea. First, banning them isn't going to get rid of the crappy posts. They'll still come in and need cleanup. I posted riddles because riddles are what I can do. I didn't think to look for a list of acceptable/unacceptable puzzle types and I don't think most people will either. The site's name is Puzzles, not "Some Puzzles Only™." Removing riddles is just going to get rid of the good riddles, riddles that bring an entirely separate intellectual crowd to the site.
I work with math all day on the job. I've got too many math puzzles already. Others simply hate math or lack the skill to enjoy them. Without linguistic puzzles like riddles, this tiny, tiny community is cutting off what I would surmise is its largest potential following. We need riddlers to get out of beta.
I propose two guidelines to keep all puzzles (including riddles) high quality.
First, that brevity is a virtue. Puzzles should be as concise as possible, with length being acceptable where required to exponentially increase the user satisfaction of the puzzle.
Second, that all puzzles fall into three categories, calculation, interpretation, and assumption.
The first category consists only of puzzles with nothing left to interpretation. There is exactly one answer and there can only ever be one right answer. The author is not needed to confirm it, because the answer is obviously the only possible solution once discovered. Even if there are a small number of equally valid calculated answers, it's okay if they're a mark of the answerers' ingenuity rather than the puzzle's poor quality. I don't see many riddles if any at all falling into this category, for they're more likely math/science related. The writers of these puzzles should have the greatest freedom.
The next category is interpretation. People are often going to guess this kind of question wrong, but sometimes in a way that is constructive and fun. A great incorrect answer will fit the majority of criterion, but never all. The correct answer will prove to perfectly fit the criterion presented with few red herrings if any. Good riddles will almost always fall into this category. Interpretation puzzlers should be required to either add a hint to their puzzle, answer it, or confirm an answer right within eight days. Moderators should perhaps warn the puzzler a day before deletion? Unanswered interpretation puzzles that are abandoned should be deleted.
The final category, assumption, can be defined as a puzzle with a low clue to crap ratio or a puzzle whose clues do not lead to a definite answer. Perhaps it could also be defined as a puzzle whose intended answer is not as good as its received answer. These are not puzzles; they are either trollish exercises in self-indulgence or works of people who mean well without the required skill, dedication, or help from the community.
Examples of Calculation Questions:
Secret Admirer Secret Message
What's the Password, again?
Examples of Interpretation Questions (yes, they're both mine to speed research time):
Internship Available!* - Figure out what you're being asked to do before you sign up
Speak the Name of the Father! - Name the lord of the gods so that he may yet survive the theomachy
Examples of Assumption Questions:
Mysterious Murder Mystery
The old man doesn't like seagull meat
To note on the last group, the murder mystery is simply frustrating nonsense. There is as much of a logical path there as there is from asparagus to KOI-3010.01. The seagull one is fun, but it's less a riddle than a story that makes you guess the ending. If it's that long, it should be a more complex puzzle that requires the length.
In summary, riddles are not the problem, they're a solution to a lot of the site's struggles. Not only will they bring new people in, the terrible riddles are great practice for moderators!