How to connect a 4x4 keypad without a microcontroller
You could just emulate what a micro would do, using logic gates. For example use a 2 bit counter/decoder and clock to select a column (1 of 4) and read the 4 rows in parallel. 

If only one key will ever be pressed at once (or at least only keys from a single column at once) then this will work without a diode per key.  A micro has the same limitations. 

It's better to use open drain drivers (and pull-up or pull-down resistors on the inputs) or at least add resistors to the driven lines so that push-pull outputs are not shorted together if multiple keys are pressed at once.