Alkaline batteries have higher current densities and are less likely to leak than other disposable batteries. They probably don't suggest rechargables because they don't output the power and hence results in a lower brightness. Also, rechargable batteries (lead acid excluded of course!) generally don't cope with higher current demands well (OK, these things are relative, but a normal bulb drains a lot more juice than most battery driven components), so you get a dull beam that doesn't last too long - not exactly what they are trying to sell.
There is no technical reason not to use non-alkaline batteries I can think of.