Here are a few sample implementations.

Place this shortcode in your page contents For This Result 
[json src="http://example.com/my_data_src?format=json" key="Data.mykey"]

Outputs contents of `mykey` in the `Data` object

 
[json src="http://example.com/my_data_src?format=json"]
I want my value to appear right here {Data.mysubdata.otherKey} in the 
middle of my content.
[/json]

Replaces the text in `{}` with contents of `otherKey` in the `mysubdata` object in the `Data` object

 
[json src="http://example.com/my_data_array_src?format=json"]
I want my value to repeat here {myKey} for every element of the array. 

I also want to repeat nested arrays: 
{array:myArray}
Repeated element of myArray: {repeatedElement}
{/array:myArray}
[/json]
 Repeats myKey for every element in the root array of returned data and repeats repeatedElement for every element of myArray JSON array property a JSON objectÂ