OK, I'm trying again but with the companion replaced by a 1 solar mass star. Timestep is 32k.

Nice sim with the 5 earths .  
I ran it and noticed the following :  
 
1. For the first time a saw the moon at AU 1 turn the other way around the earth ! This happened at the longest elongation point  ( after about 6.5 years ) where the moon choosed the other side to "round" the corner . Then the cycle repeated ...
 
2. The other earths have the same phenomenon , but it takes a lot longer ....
 
3. Max eccentricity occurs at about :  
 
AU1 : 6.5 y
AU1.5 : 13 y
AU2: 28 y
AU 2.5 : 74 y  
AU 3 : 158 y  
 
plotting this into excel gives : time,max exc ~~ 1.10 exp(1.65 AU )  
 
For Uranus ( AU19.22 ) this gives : 7...e+13 years !! Longer than the lifetime of the universe  
 
please note that the red simulations were done at higher time step .  
 
Things can be optimized but above indicates that the Kozai is surely  dependent upon distance  
 
Another parameter surely involved is the distance of the moon from the mother planet ...

Yes, it is gaining lots of ecc per orbit as the slope of the curve gets steep.  It's outputing the data every 10 days.
 
Since I placed them in 90 degree orbits, I would expect to get the maximum possible eccentricity, which appears to be just under 1 (0.994).  I would guess that if I placed the moons in orbits with 80 degree inclinations that max ecc would be a bit less.
 
I'm doing a new run.  I'm up to 18 years now and the ecc of Earth1's moon does indeed drop back down almost to zero before rising again to max ecc of almost 1.  Earth2's moon is starting to climb on the graph.  I'm guessing that it too will approach 1.  If so, it sounds like that conclusion is correct, that the max ecc will be the same, but the period of the oscillation will differ, and as this illustrates, by quite a bit.  I'll post it later tonight.  Maybe Earth3's moon will be making a little noise by then too.

A close planet is probably better protected than a distant one.  Like Frank said, the distance probably matters a great deal.  The Earth's moon is not buried deep within Earth's Hill Sphere.  Rather, its 1/3 of the way to the edge.  This gives us an excuse for another simulation!
 
I'm trying the 5-Earth sim over again at a timestep of 4 since 16 is a bit fast for the close approaches at max eccentricity.  I want to see if I get the same numbers as Frank.
 
I would guess that the quickest way to get the numbers on a 500 AU system would be to start with the companion only a few AU away, record the time to max ecc, then move the companion out a few AU, record... and plot a graph, extrapolating out to 500 AU.  It'll probably be similar to Frank's conclusion about Uranus' moons being protected from Kozai for timescales that exceed the life of the universe.

Quote from Mal   on 10/24/06 at 08:10:53:

 
Home now. Here's the results so far... the eccentricity of the planet is varying on a regular pattern on a 400000 year cycle, but it's only varying between 0.63 and 0.58 (initial ecc was 0.60). Planet inclination has increased monotonically from 0 degrees to about 12.8 degrees over time (this is over 6.7 million years).
 
So it looks like increasing the companion mass makes the planetary inclination rise faster, unless that's down to the shorter timestep?  
 
I'll start this again with the numbers in the paper and see what happens - I think I can afford a 65k timestep here since the planet's at 2.5 AU there. The inclination increase is strange though - the eccentricity isn't doing anything at all really.

I've just finished the five earths sim up to max ecc for the fifth earth . Seams interesting . I will send the data as a plot of time tomorrow ...