Japanese spelling was drastically reformed in the period following the second world war. Prior to this, Japanese spelling (in kana) was quite irregular. There were several ways of spelling things which were in fact all pronounced the same way.
For example, ikimashou was written as ikimaseu, the sokuon or "small tsu" character was written as a large tsu, the verb iru was written as wiru, and so forth. People didn't speak like that: they spoke as they do now, but they wrote like that.
In the period of spelling reform, most of this old stuff was swept away and replaced with the modern system, which is more phonetic. For example the kana for we (ゑ) and wi (ゐ) are no longer used in modern Japanese because these sounds have disappeared. This meant for instance that several old kana writings were mapped onto one new one. For example both kuwa and ka were mapped onto ka. So, it's relatively easy to make a set of rules for making new spellings from the old ones, but one cannot work out the old spelling from the current one.
You can find a complete list of the different types of old spelling, and which new sound they are mapped onto, in one of the appendices of the Kōjien dictionary. You'll also find the old-style spelling in this book, and in almost any other Japanese dictionary.
What irregular spelling is left over from this spelling reform? Basically, the following:
The particles pronounced as on the left are written in kana as on the right.
Tsu and chi syllables (also known as moras) that are nigoried (have the `` marks on them) are pronounced zu and ji, exactly the same as ず and じ. Most of the old spellings which used づ, (which we'll call du), and ぢ (which we'll call di), turned into ず (which we'll call zu) and じ which we'll call zi.
Hepburn and Kunrei romanization map both the sounds tu'' and su'' and ti'' and si'' onto the same romanized letters, because they are the same sound, but it makes converting back to kana difficult. Here we use nihon-shiki romanization.
For instance itadura, `mischief', became itazura, which is pronounced the same way. What did not change, were the words which clearly originated from a compound where the tsu or chi of one of the compound words had been nigoried in the process of compounding. For instance, kanadukai, meaning `kana usage' from kana and tsukai, did not change its spelling. Again, a list of the words which did and did not change their spelling in this way can be found in the Kōjien dictionary.
In the chōon (lengthened vowels) of Chinese-derived words such as kōshin (renewal), the u kana is actually pronounced as a continuation of the "ko", so in fact it's an "o". However, this does not apply to all words: the u of omou is not pronounced as an o, it's a u.
Similarly, the combination of kana ei is usually pronounced as ee. This is sometimes relfected in spelling. For example, keitai, mobile phone, is often written as ケータイ in katakana. However, some words, such as tameiki "sigh", meiru "be depressed", or ei "ray" (the fish) are pronounced as ei, not ee.
There are four different pronunciations of the kana n"depending on the sound that comes after it.
There are two different pronunciations of the consonant "g" (the gagyō, ga, gi, gu, ge, go)):
Kana does not show the pitch accent of the words.
The verb iu (to say) is often pronounced yū.
There are at least two ways each to read the kana ki, ku, shi, su, chi, tsu, hi, and fu, because the vowels may or may not be devoiced.