First of all, I like the title, "Cry You Rivers." It reminded me of the title's song "Cry me a river" by Justin Timberlake.

Second, I just want to let you know that the following comment is my opinion, and that I may offer some suggestions but that you are more than welcome to either use them or to take them into consideration for future poems.

With that being said, I'll shall continue onto my comment.

"You can lie to me,
I actually look forward to your lies.
For right through them I can see,
The give away are your eyes."

-- I can see you like rhymes. I feel that by focusing too much on rhymes, it may hinder the flow of yours words though not of the emotion. So I'm sure that eventually as you continue to practice your writing, you'll figure out how to write with rhymes without making some lines within your poems sound a tad force. By a tad force... I mean on the third line, it sounds force. The reason why I feel it sounds force is because in the first lines of your stanza, your syntax is arranged in a "normal way of speaking." However, if you noticed, the third line in this stanza, differs from the syntax of your first two lines.

"for right through them I can see"

^
"I can see right through them," this is how people would normally say it. But if you write it this way, it would not go along with the rhyme scheme you are using. So perhaps, you can experiment with that line? Perhaps, you could change the rhyme scheme? I mean, you don't have to change everything in this stanza. May be, just a few words?

"the give away are your eyes."

Now on this line, "give away" should be together. At least in this sentence, I think "give away" is working as a noun rather than as a verb, therefore, it should be "giveaway". And since, "giveaway" is singular, it should be "is" instead of "are." Like this: " the giveaway is your eyes."

or "your eyes are the giveaway.

"And I would go away,
from this beautiful land of yours.
And I would go away,
taking away my remorse.

But remember, this one tiny thing,
there will come a day when the birds won't sing.
When the forests would be bald and bare,
Nothing at all, the storm would spare."

-- These are my favorite stanzas of this poem. They sound smooth, though, I think the last line would sound better as:

"There will be nothing the storm would spare"

"So come to me that day,
when the drought leaves you in slivers.
So come to me that day,
and I'll cry you rivers."

^ I like the ending. I wish I could give you my interpretation of this poem, but to be honest, I'm feeling tired at the time. I might come back tomorrow and explain what I think the meaning of this poem is.

Keep writing.