The deal with tree roots is that they generally go out to the end of the dripline of the leaves. The best way to get rid of the roots is to make sure first that the driplines don't cross over any of your sprinkler system.
Another truth about the tree roots is they probably are all about the same depth.
You basically have about 4 options.
1. Do nothing, incremental repairs to the system as problems occur, or close off that section of your system and use a hose sprinkler.
2. A minimal approach. Make changes to the system that keep it away from the trees dripline.
3. A more aggressive approach. Make changes to the trees dripline, and/or the system to keep them from crossing into each other, including perhaps digging a main line down to a deeper level.
4. A radical approach. Kill the neighbors tree and end up with a neighbor that likes to burn out your lawn once in a while with some simple road salt. If my tree got killed by an unruly neighbor, I'd be salting the guy late at night on a pretty regular basis or just once during the winter when its impossible to see, and it's pretty hard to prove, so I'd recommend being very neighborly about it all. He, in fact, might not mind if you cut some of the long branches that overhang your yard (assumed), off to a shorter length, and this is your legal right in most locations if he is disagreeable.