Good question. I don't know since I don't know exactly why linear regulators have stability problems in the first place. That said, check out the REG101, 102, and 104 regulators from TI. THey are LDOs and have adjustable versions, and from the internal block diagram they work the same way as the LM317, except that they require a ground pin. So they probably work as current sources too.
DId you notice that the LM317 DOES NOT need capacitors for stability? Just letting you know...that might be why they caps aren't needed for the current source. I don't go around building current sources so I do not know if anyone has ever used a linear regulator that needs caps as a current source. THe one time I needed a current source, I found the LM317 and looked no further.
EDIT: It also seems that the LM338K also needs no capacitor. It might be that the design of linear regulators which allows them to be wired as current sources inherently do not need capacitors. Which LDO did you have in mind?