I tried googling about laptop adapters being used as LED drivers. I tried to to use it and it worked fine. Some articles says that laptop adapters are voltage-regulated and not constant-current sources, so they shouldn't be used.
I am trying to make exactly what is shown in the picture. I want to know first of all if a laptop adapter can be used, and if this will work in the long run with a heat sink.
Due to the fact that most of the laptop adapters are constant-voltage (i.e. voltage-regulated) converters having very low output impedance, you'll need series resistors for each branch to drive the LEDs properly and to protect them, as shown in the image. It's practically applicable.
There are possible issues, however:
Make sure the output voltage rises from zero to its nominal value (~19V) properly at turn on (i.e. when the line is applied to the charger). If it jumps to a higher voltage at turn on (overshoot) and if the amount of overshoot is substantial then the LEDs may get damaged (maybe a low chance but still, nonzero). Since the battery charging circuits inside the laptops can generally withstand these overshoots, the adapter designers may not have gone for a low-overshoot necessarily.
Most of the adapters are power-factor-corrected. For simplicity and cost effectiveness the designers prefer PFC-combos and this approach causes a substantial 100 Hz ripple at output (This is not an issue for laptop battery charging circuits therefore the designers may not bother with reducing or eliminating the ripple). So the LEDs will probably have flickering. Although the flickering won't be noticed with naked eye, it can cause an eye tiredness over time. NOTE: I don't know the application of your LED lighting as it's not given but if it's going to be used as, say, a room lighting then working in that room for a long time may cause eye tiredness or even headache.
PS: I normally wouldn't prefer these adapters for such purposes. Instead, I'd personally prefer a post regulator to drive the LEDs so that anything may possibly come from the adapter don't harm the LEDs.