It is so called the preposition-LESS  accusative. It indicates the continuing period during which last the action.

One of the tricks of Russian is to denote time by accusative:  

Поезд отправится в пять часов.

Буду там ровно в шесть. 

Он выехал в ночь. 

"В пять, в шесть, в ночь" - accusative with preposition.

Каждую ночь - accusative with no preposition, but the action lasts.

Accusative may be used in showing directions, but here it is like the "timing direction". "Я иду в дом" - this is the "space" direction.

"В прошлую зиму было много снега" - this is the "time" direction.

Hechmi, oh, using cases with time units is a bit complicated matter:)

It is often accusative, sometimes prepositional, sometimes locative, sometimes even instrumental. Mostly depends on the word used, but some allow more then one case (with mostly stylistical difference).

With каждый we use accusative.... normally.

Nominative is possible only when 'night' is the subject, like "Night is a beautiful time", but when you speak about something happening "in the night" - no, it is NEVER nominative.

Каждую ночь

Каждый день

Каждую неделю

Каждый месяц

Каждый год

These are fixed expressions use with the accusative case. You just need to remember it :).

Now about 'why'. Normally referring to location of anything we use:

в (in) or на (on) + prepositional/locative case.

When we speak about the direction of movement, that is when Russian в/на = to, into/onto, we use в/на + accusative.

Also, in past Russian found (and still finds to lesser extent today) use of instumental to mean 'mode/manner of action'. To answer 'how'. Sometimes the place where you do somehting can be put in instrumental too. E.g.
<em>XXX шёл лесом</em>. - literally "XXX went by forest". (иди лесом also serves as euphemistic for the Russian version of 'fuck off')

So I don't know, why did you expect nominative here! You should expect prepositional:)
But for some reason most of time units come with accusative instead of prepositional:-/ And this is really old thing, so I don't know why.
So of them are used with prepositional too, and for many of them instrumental is also popular.

Thank you for the answer guys. 

So it's just one exception I guess right? Actually, it would help me a lot if one of you gave me examples of Каждую ночь и Каждая ночь in sentences.

"Каждую ночь "- in the sentence answer on question - " WHEN ? " 

Каждую ночь я вижу сны - Every night I dream. -   in this sentence the subject that do the action is - I.

"Каждая ночь" - in the sentence answer on question - " WHAT/ WHO ? "

Каждая ночь приносит мне новые сны - Each night brings me new dreams -  in this sentence the subject that do the action is - night .