Many good posts concerning this matter, I found the answer finally in the Appendix A-F from "The Return of the King". And it's much in conjuction with what "Saucepan Man" already suggested.
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And quite possibly Dwarves were immune to that aspect of the Rings' powers, in the same way that they were not susceptible to the power of the Rings to draw the bearer into the realm of shadow.
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I'll translate what I found freely, "..the only power the ring had over them was that it poisoned their hearts with an overmastering greed for gold and valuables".
Later it states "They could never be turned to shadows under someone elses will.. nor had it any effect upon their lifespan, so that it made them live shorter or longer". Since the invisibility obviously was connected to the entering of the wraith / shadow world they probably didn't have this possibility either.
I guess this quote wouldn't apply to dwarves then..
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According to Letter #131, the Seven and Nine conferred invisibility to the user as well as unending life. Also, the ability to make the wearer invisible seems to be a controllable power - not always "activated", but rather selectable when the wearer desires to be invisible. However, eventually the user would fade and become a wraith under the control of Sauron, the Dark Lord. The Three Elven Rings did not confer invisibility, because this is a power from Sauron, and Sauron did not have a part in the creation of the Three Elven Rings, nor did he ever touch them.
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[ July 03, 2003: Message edited by: Gorthol ]