Switching Circuit Using 4017
15ms is pretty slow in microcontroller-land, so an arduino would be a perfectly-acceptable solution.  Certainly the easiest to adjust and extend.

If you want to do it in discrete logic though, I would suggest that the important part you want is a 4-bit serial-in/parallel-out shift register, and not a counter. 

Something a bit like this:







simulate this circuit &ndash; Schematic created using CircuitLab

Edit: I've just noticed what a 4017 is.  It would work with the AND array, but you need a separate RESET input so that its sequence goes back to output A after 4 pulses, otherwise you will have 6 no-output pulses between D and A.  The NOR arrangement in my schematic means that you will always have exactly one active output and it will cycle through A-D.