The Austronesian language of the Marshall Islands in the Pacific. It is noteworthy for several versions of its alphabet, having a rather peculiar set of accented letters. This apparently corresponds to a distinctly peculiar arrangement of sounds. What follows next is the description I've read most recently: this is in an article1 in a phonetics journal, so it must be accurate. After that comes the wreckage of earlier attempts I made to describe confusing transcriptions I'd seen.

It's a strange system by world standards: Marshallese has three sets of consonants, palatalized, velarized, and velarized rounded. These can be labelled with the features [−back, −round], [+back, −round], and [+back, +round] respectively. Here I'll write them as e.g. pj, pg, and pw respectively.

Its vowels are at four heights, and it has front vowels, back unrounded vowels, and back rounded vowels roughly at each height. That is, using SAMPA symbols, it has:

        i    M    u
        I    7    U
        e    V    o
        E    6    O
But these vowels aren't all contrastive. In fact there are only four contrastive vowels, the four heights. These may be described as [+high, +ATR], [+high, −ATR], [−high, +ATR], and [−high, −ATR] respectively. Their exact quality in terms of [round] and [back] is determined by the surrounding consonants. (What, no CV or VC syllables? I didn't see what happened to them.) So between two palatalized consonants the highest vowel is pronounced [i] but between two velarized rounded ones it's [u]. Between two different ones it's a continuous transition, as in [pjiukw].
And now for some other, less accurate, descriptions. The following are orthographies attempting to represent the above arrangement. I haven't updated it to take account the true nature of the sounds (above), but am letting it stand as testament to the problem of phonemic spelling:

Here is one system I've seen. M and N can have a single vertical stroke above them, G N L and R can have double vertical strokes, and L can also be slashed as in Polish. I have no information on how these are pronounced. The ampersand '&' is used as a vowel, said to be intermediate between E and I.

The actual Marshallese names of some of their places in this transcription is interesting, but doesn't help much at guessing at the pronunciation. I am using M' for M-stroke and L/ for L-slash:

M'ahjel/ = Marshall Islands
Majr&w = Majuro, the capital
Kiwajleyen = Kwajalein atoll
Pikinniy = Bikini atoll
Yepjay = Ebeye atoll
Yan&yweytak = Eniwetok atoll

And some borrowings from English are:

baham' = bomb
Hamedkah = America
tiraj teyr&yt&wr&y = trust territory

This is not the only transcription used. Another script (which might be somewhat sanitized since it was being used on the Web) has these forms:

ri-Majöl = Marshallese
kajin Majöl = Marshallese language
kajin Pälle = English language
aelön = atoll
yokwe = hello, goodbye
aet = yes
jaab = no

The numerals from one to ten are:

juon, ruo, jilu, emen, lalim, jiljino, jiljimjuon, rualitok, ruatimjuon, jonoul

tres equis has found a sample with yet another system:
http://www.udayton.edu/mary/images/marshall.jpg
The accented letters in this are A-macron, O-macron, and N-macron; and M-cedilla and O-cedilla. Again, no information on pronunciation. -- See below for details. -- One grammar I found seems to say there are two competing systems, but it was not entirely clear. *sigh*

I've been peeking into a dictionary in a bookshop, and it seems some of the accented consonants (however denoted) are velarized. The J is a palatal stop. N-macron is a velar nasal. There are front and back A, and a central vowel. There is also a bit of dialectal variation.

1. Mark Hale, 'Marshallese phonology, the phonetics-phonology interface, and historical linguistics' (2000), The Linguistic Review 17: pp. 241-57.