Korean Hangul and Japanese Kana may be some of those.You may want to check this up: http://www.linguistics.ucla.edu/people/schuh/lx001/Discussion/d04_phonetics_key.pdf
Yes, Arabic and Hebrew script can be considered to be syllabic writing system that does not take vowels into account. At least from the historical perspective.
If you're referring to cases where one glyph represents one syllable then for partially syllabic writing systems Japanese and Korean are, as Adria Rofes said, good examples. As is well known, Japanese is written with a mixture of Chinese characters (Kanji) which form the lexical base and hiragana used primarily for grammatical markers, and also katakana to phonetically represent foreign loanwords (I'm simplifying somewhat.). Japanese is partially syllabic in that while the katakana and hiragana syllabaries are made up of one symbol to one syllable the individual Kanji may be multi-syllabic in their expression.
Korean, on the other hand, is partially syllabic for a different reason. This also has a vocabulary mix of native Korean, Sino-Korean and an increasing number of foreign imports largely from English. Up until recently, mixed script - like the Japanese - was common in many publications, but nowadays almost exclusive the native hangul is employed to write Korean and transcribe loanwords. Hangul is an alphabet but, uniquely, is not written linearly, but with letters stacked-and-packed in a sequence of syllabographs, with up to four letters in one syllable block. It is thus an alphabetic syllabary.
A good recent reference is Hye K Pae (2011) in Writing Systems Research 3:2, 103-115.
Your answer for this question is appreciated. I can understand from your reply that Koran, Chinese and Japanese languages use a partially syllabic writing system. Thank you for the reference you have suggested.
You will find plenty of nice examples here (the Ge'ez writing system is used for Amharic, mentioned above):
http://www.omniglot.com/writing/syllabic.htm
http://www.omniglot.com/writing/syllabaries.htm
See also
http://en.wikipedia.org/wiki/Syllabary
Note the subtle difference between a purely syllabic script and a "syllabic alphabet" (though the syllable is the main unit in both, and the difference is often ignored in the literature). In some languages (e.g. Chinese) words are normally monosyllabic, so if a language like that uses a logographic script (whose symbols are meant to represent whole words as units of meaning), such a writing system is also effectively (if accidentally) syllabic.
All languages in the Mayan group are combined Syllabic and logographic (sorry Sir Eric). In written form, you might not consider Mayan a modern language group but it is still spoken throughout Central America.
For those interested in more information on the Mayan language group in general and also in its written form, I recommend "Reading the Maya Glyphs" by Michael D. Coe and Mark van Stone. As I've not these scholar's understanding, I yield to their own words and offer these two snippets for you to test your interest against.
"All of the 31 extant members of the Mayan language family, along with Classic Mayan, are grammatically very different from the languages (all belonging to the Indo-European family) that we either know as native speakers, or from having studied them in school.
"Maya writing has both a semantic dimension and a phonetic one: some signs indicate meaning alone, while others express particular sounds in the language being recorded. In this it is similar to other ancient scripts, such as Egyptian, Chinese, and the cuneiform system of Mesopotamia. We call the "meaning" signs logograms; these stand for whole words or word-stems. For a list of the most common logograms in use among the Classic Maya, see pages 161-166. The phonetic signs are syllabograms, representing syllables (combinations of consonants and vowels), as well as the "pure vowels" (a, e, i, 0, u) unaccompanied by any consonant. Maya writing is thus logosyllaoic. It is no accident that there is no known script in the world, ancient or modern, which entirely consists of logograms - there would simply be far too many discrete signs for anyone to memorize, and too much ambiguity inherent to such a system. Accordingly, the scribes of these early civililizations attached signs known as phonetic complements (PC) to logograms (L) to help in their reading. Maya syllabograms were used in just such a way, as we shall see.
"Yet, in the Maya system, any word that could be spelled with a logogram alone, or by a logogram with one or more phonetic complements, could and often was written purely syllabically (we shall examine the rules for this in 2.2). So why didn't they give up the clumsy logograms, and write everything phonetically with syllabograms? The probably answer is that the Maya elite, including the scribes, put a high value on signs which not only had deep cultural meanings, but which could be relatively difficult to write (at least for neophytes)."
Chinese written with Chinese characters uses a(n almost 100%) syllabic writing system. Japanese kana, although aften referred to as syllabic, are actually moraic, so a syllable like hon or sai is written with two graphs, one for each mora. The same probably goes for any language which uses what is called a "syllabary" -- other examples are Mycaenian Greek written with the Linear B script, and Hittite Cuneiform (where, for example, the city name Kiskilussa is written as ki-is-ki-lu-us-sa, i.e. 6 graphs for 6 moras (but 4 syllables)).