Is it possible to just take a normal (continuous) laser diode and pulse it? Or does this not produce the same effect as pulsed laser diodes?
Often, a pulsed laser diode is just one that is not very efficient, so that if it were operated continuously (we say "continuous wave" or "CW") it would overheat and destroy itself. Most often the first lasers designed of any particular type (including the first laser ever, a pulsed ruby laser) are pulsed, and only later are more efficient ones capable of CW designed. This includes laser diodes.

So you can certainly take any laser diode and operate it pulsed.

Doing it practically will have a few difficulties.


If you want to achieve very short pulses, and the laser isn't designed for that (very low capacitance, mainly, when we talk about laser diodes), then you might not be able to get the pulse width you want with any particular laser.
If the laser isn't characterized for pulsed operation, you won't know how strong a pulse (what pulse energy) you can drive without damaging the device or reducing its lifetime. 
If you want a very particular laser wavelength, you might find the laser produces a slightly different one when operated pulsed than it does when operated CW.


Also, if you want pulses in the 1-100 ns range, and not too much peak power, and are willing to do some experimentation (and destroy a few lasers) to find how hard you can drive your laser without damage, then you will probably be able to do it. If you want sub-nanosecond pulses (and particularly if you want 10 ps or less), you'll probably want to choose a different laser type with a gain-switching or mode-locking mechanism and a longer cavity to achieve that. Diode-pumped fiber ring lasers seem to be quite popular for producing sub-picosecond pulses, for example, depending on the desired wavelength.