Controlling the centre wouldn't be as important.
What would chess be like without the 4 central squares??

Would the game and strategy be drasticallly different?
No, the game and strategy would be exactly the same. You wouldn't notice any difference.
How would the Bishops get across the board if there were no middle squares? Come to think of it how would the other pieces be able get past the middle? What would you have insead of the four centre squares? A sort of no go area that all the pieces had to negotiate their way around? Or just one big square, like an English road 'roundabout'.

You seem to be asking "what would chess be like if it wasn't chess"?
Yea! be sensible.

Too drawish. I think it would be too easy to fortress on 3 files. Without the central squares the ability to pressure the flanks is greatly diminished. For the same reason organizing a defense / fortress will take much less time than organizing an attack. Also attacks in general will lack punch as the board isn't as dynamic in terms of mobility.
What would chess be like if squares e4,e5,d4,d5 does not exist
To make you guys understand it better
Ex. Ba1 can't go to h8, only to Bb2 and Bc3
Rd1 can't go to Rd6, Ra4 can't go to Rf4 etc
d and e pawns still exist, but they can only go up to d3,e3,d6,e6 unless a capture takes place like exf4
they can still promote normally by going to the final rank (Ex e2-e3-exf4-f5-fxe6-e7-e8=Q)
Central pawns on the third rank will only be able to control one square ( Ex. d3 only controls c4 since e4 does not exist/ e4 is a empty space)
Would the game and strategy be drasticallly different?