Is the following phrase grammatically correct? 
     'Inertia Dispatches All Hope Obediently'.
(It has to do with an article relating to 'Idaho')
Thsnk you.

Did you read this, or is it your own text?

Except fot the capital letters on the words following "Inertia", I
would argue that it is grammatically correct, but the sentence does
not make any sense.

The grammatical form of the words in order is:  Subject(noun), verb,
adjective, object(noun), adverb.

This is a typical sentence structure.
Example:  Water dissolves some salts immediately.

(The adverb "immediately" could also be placed before or after
"dissolves" without changing the meaning of the example.)

Now let's see what others have to say.

Yes, it's grammatical, as is the famous sentence "Colorless green
ideas sleep furiously."

http://en.wikipedia.org/wiki/Colorless_green_ideas_sleep_furiously.

I think, not only the sentence is grammatically correct but also makes
perfect sense.

It's crackers to slip a rozzer the dropsky in snide.