Translation:My grandmother gets up early every morning.
I'm not sure if there's a reason behind this rule (maybe someone else has gone deeper into the grammar than I have), but when you use a specific time without a preposition, it's always in the accusative case. This page gives some more examples: http://goo.gl/IFthZL (you'll have to scroll down a little to the section "Some rules for adverbial time phrases")
[EDIT: See below and other explanations - don't confuse separable verbs with prepositions! Just memorise that most time expressions use accusative.]
[Old comment:] Actually, I understood it to be because of the auf, which is part of the separable verb aufstehen ("to get up", notably different to aufwachen, "to wake up (but maybe still lie in bed)"). Auf is a two-way preposition, which means it forces dative case ("jeden Morgen") if it's used for motion, which "getting up" is. This is the kind of thing I'd like clarification on, though.
Separable parts of verbs are not real prepositions, they don't use any nouns. There seems to be a simple rule saying that whenever you name time when actions of your sentence are happening, and you don't use prepositions, you put the time in accusative. Similar rules probably exist in any language with declension (I'm sure for polish, latin and ancient greek) and are main reason for having it.
Your English is not technically wrong, but placing the adverb "early" at the end of the sentence is an unusual construct. One normally places the adverb as close as possible to the verb it is modifying: ". . . gets up early . . . ."
I don't know if there's a particular rule regarding whether the adverb goes before or after the verb. For example, I said "One normally places the adverb . . . ." You could say "One places normally the adverb . . . ." but that would sound awkward and stilted.
The payoff is that the flexibility makes poetry easier:
"My grandmother is such a girlie // She gets up each morning early."
Native English speaker here: let's keep in mind that we're looking for things that are or aren't grammatical. To me at least, "My grandmother is getting up early every morning" is grammatical; depending on this situation, it could be the best sentence to use. E.g. "Now that she goes to bed earlier, my grandmother is getting up early every morning" or something like that.
Thank you all for the further clarification. I agree with your comments. I almost did not leave my original comment because I did not think dmarcovic's translation was "wrong". However I also thought it did not sound right in this case. Upon reflection that is because when presented with only one sentence and no context I had put it in context for myself, but as you said there could be other contexts in which it is more appropriate.
I likewise don't know what you'd call it, but to me (NAE) ". . . is getting up early every morning . . ." has a sense of doing something now, and continuing to do it. The construct " . . . gets up early every morning . . ." implies a past aspect as well: my grandmother has, is, and will get up early.
EDIT: There's some misunderstanding here. I think dmarcovic's translation is valid and should be acceptable. That said, there is a nuance of difference between the two solutions.
When we say "every morning", we are talking about something that happens regularly. Hence a simple present tense should be used, i.e. "gets up early". "Is getting up" implies a present continuous tense, which is used only when describing things which are particularly ongoing at the moment.
There are more possibilities than just those. I refer you to the following, excellent article on German work order: http://www.dartmouth.edu/~german/Grammatik/WordOrder/WordOrder.html
meine = my
Großmutter = grandmother
steht . . . auf (infinitive: "aufstehen") = gets up
jeden = each
Morgen = morning
früh = early
So, preserving the German word order, we have, "My grandmother gets every morning early up." With standard English word order that reads, "My grandmother gets up early every morning."
As for your second question, "aufstehen" can mean either "to stand up" or "to get up". http://dict.leo.org/ende/index_en.html#/search=aufstehen=0=basic=on
If you want to know more about German word order, I highly recommend the following article: http://www.dartmouth.edu/~german/Grammatik/WordOrder/WordOrder.html
Placing früh right after steht may be acceptable/natural. It may also be that having it right before the separable portion auf is essentially the same thing.
On the other hand, does it not to some degree also modify Morgen? What we have here are two modifiers describing when die Großmutter arises, and both would compete for that first position used for temporal.
so i put wakes up instead of gets up why is it wrong.
Because those are two different things.
Waking up (aufwachen) is to resume consciousness after your sleep.
Then you open your eyes.
You might still lie in bed for ten minutes before you actually get up (aufstehen) and put your feet onto the floor.
morgen früh is an expression that means "tomorrow morning".
This sentence uses jeden Morgen (every morning) and then uses the adverb früh (early) to describe the action aufstehen (get up).
The fact that the word Morgen is next to früh in this sentence is a coincidence. The two words do not belong together here.
You could also say, for example, Meine Großmutter steht jeden Morgen sehr früh auf "My grandmother gets up very early every morning", with sehr früh "very early".