What are the limitations on crossword puzzles in other languages that use phonetic alphabets? Italian comes to mind as a language in which they would be impossible, because every Italian word ends with a vowel, so there could be no consonants in words along the right or bottom margins.
Would Scrabble games have similar issues in some languages?
Not uncommon for crosswords to use abbreviations, foreign words, slang… Probably could work around all words ending with a vowel? Or use crosswords that look like extended tic tac toe grids and not groups of square boxes.
The problem you refer to is only severe with American-style grids. Most other countries — including England! — use one of several different grid styles, in which a letter may be in a single word instead of two.
Crosswords here in Italy are very possible, here’s an example.
Short words usually are acronyms or some other wordplay (like “the end of the tale” would give LE)
We have two different version of Scrabble, and both are popular.
I’m wondering about languages, like Hebrew, in which vowels are customarily omitted. If a letter has an implied vowel in one direction, would it necessarily have the same implied vowel in the other direction?
Hebrew has crosswords just like English. Your question about the implied vowels doesn’t really matter…The written language uses only consonants.
For example, lets take 2 Hebrew words you already know :Halleluyah, and Shalom.
They are spelled in Hebrew like this : HLLYH and SHLM
So in a crossword puzzle you could write them just like in an English -language crossword:
S
HLLYH
L
M
(disclaimer, for folks who know some Hebrew: I have very over-simplified this example, for educational purposes. Even though we’re Dopers, let’s not go nitpicking it over the use of the letter Vav as a vowel, etc. Okay? 
)
Oh, that’s pretty interesting. It doesn’t quite follow the rules of an American-style crossword (each letter must be part of another word, so no dangling single letters, and words must be at least three letters long–although that particular crossword does seem to follow that part) but it’s pretty impressive. I was more used to seeing the lattice-style crosswords throughout Europe, with occasional “small blocks” like this in Polish, and I’m pretty sure I’ve seen the exact same type in Hungarian, but never a full American-style grid.
Some preliminary Google image searching disappoints me in not turning up crosswords done in Egyptian or Mayan glyphs. 
I think a more serious problem would occur in a logographic language like Chinese, where each word is written as one or two (rarely more than two) characters, leaving little room for including characters in more than one direction. Nevertheless I see from Google that there are apparently crossword puzzles in Chinese. However the few which look similar to English designs, like this one, cheat by using numbers rather than actual words. Names of numbers in Chinese can be longer, like English “forty two thousand five hundred sixty four” which in Chinese would be the nine character sequence 四万二千五百六十四.
This essay in Quora may also be of interest.
And a question about languages with accents, say, French. If a vowel is, say, an E with one kind of accent for the across word, does it have the exact same accent for the down word?
The other way they do it is for the answers to be compound words or phrases of 4 or more characters, which you can also see googling for images of ‘Chinese crossword puzzle’.
In Korean it’s similar. Although there’s an alphabet, letters are grouped into blocks of 2-4 which form phonemes representing either the Korean sound of a Chinese character in typically two syllable Sino-Korean nouns, or one or two syllable indigenous nouns (verbs might be longer) or the syllables of non-Chinese words of foreign origin (mainly but not exclusively from English). Crossword puzzles have one phoneme per square not one letter, with compound words/phrases or foreign words used to help generate enough entries >2 squares. It’s not that you can’t make one, but they tend to end up pretty simple, and seem to be oriented mainly toward those learning the language.
In general I guess this would be the answer for any language with a complete writing system of its own: that you could make some kind of crossword puzzle, but not all languages and writing systems lend themselves to it to the same degree.
Can we nitpick about the fact that SH would be one letter? 
Accents are usually not taken into account, it’s still considered the same letter.
Other cases such as ñ or å, which are considered letters in and of themselves, are taken into account.
I think that extends to languages with short words… monosyllabilic languages.
What happens is that modifies on each character in the logographic language actually adds an word. Dry plain = desert. Icy plain = arctic.
So everything is two or three words.
Well short words are too easy. We have fun with the obscure spelling of various long words… which word has uu ? which word ends with thym ? That sort of thing. Monosyllabilic languages are just not hard or impossible.
I am informed that modern vietnamese is kind of different to the vietnamese of 1960’s. So vietnamese just arriving from vietnam and talking to the refugee vietnamese… find it hard … they use different grammer, the word pairs have changed…
Dang … I manage a large, multilingual, international website, with an Italian presence. Despite that, I did not know this about the language. Thinking you had to be incorrect, I checked my site.
Well, ignorance fought! Well done, Sir.
It’s not all the words, but the vast majority of them. The directions, for example, all end in consonants. There’s plenty of loan words, too, that do. And, of course, there’s words like “il,” “del,” “per,” etc. Of course, you can’t make a whole crossword out of those.
Can we also nitpick that written Hebrew does have a system of indicating vowels when so desired?
Japanese can be written entirely in kana syllabary, but “ン” “n” is the only consonant which can be used to end a word. They also have crossword puzzles which are done in kanji.
American crosswords can have single word letters as well, in the corners. I think German crossword grids are more like the one used in British crosswords. What I find really disconcerting about the German crosswords I do–and I have to say unaesthetic–is that you can have a sequence of two or more letters that don’t belong to any word.
AFAIK only (North?) American crosswords have the symmetrical arrangement of light and dark squares.
Maybe some variants do, but not in a standard American-style grid. The usual rule is that every letter must be a part of another clue and that word must be at least three letters long.
ETA: