use capacitor/resistor to delay the moment they turn on.
This idea in itself is not likely to work very well, if at all. I highly recommend trying to do it anyway, as experience is a valuable lesson. Basically, the LED will turn on slowly as the voltage rises, and the resistor you need to get it slow enough wouldn't pass enough current to light the LED to a useful level.
That didn't stop me from thinking about the question :)
You can make an LED in series turn off by shunting current around it. Those fancy words just mean "short-circuit the LED". As you understand the current to be the same through the series circuit, the other LED's would light up. The problem, then, is that there is no voltage across the shorted LED, so it's hard to put any kind of circuitry there.
But wait! If we put a capacitor across the LED, we start with a short--all current is going through the capacitor in order to charge it. Then, when the voltage reaches the LED's voltage, the LED turns on and the current goes through that way, even though the capacitor isn't taking any more current.
simulate this circuit – Schematic created using CircuitLab
The trick that will give this thing a chance of working is the constant current source (I1). This eliminates any issues with the fact that voltages in the circuit are all changing as it charges up. And you don't have to use any resistors to set LED currents. I chose 10mA because the actual capacitors will end up being very large, so the lower the current, the slower their voltage will rise.
I still don't think the results will be all that good but given the current source and the limiting effects of the LED's I think it actually has a better chance than the parallel version. Again I suggest trying it if only for the experience. I may even try it myself just to see!
By the way, the schematic maker wouldn't let me turn the current symbol over, so the arrow points down. So I changed it to -10mA, just because.
Oh, and you can buy things called Individually Addressable LED strips. You drive them from a microcontroller. This probably counts as "cheating", but that's for you to decide. What you get is the ability to control the color and the brightness of each and every LED on the strip!