It really isn't the worst idea for a triplex, as they can be rough the on surrounds if the operator doesn't turn carefully. Still, the direction would have to be varied as was mentioned. However, since most greens are circular or ovular (as much as we like square greens and odd shapes round here) this is no big deal.
There is no real advantage for walking greens mowers, as they can be turned in no time at all, and having the operator walk the surround from the first pass to the middle of the green would just waste time and add steps. Greens routes are already 4-5 plus miles of walking in my hood.
In one of the Courses By Country profiles on this site there's a set of greens mowed light/dark which I've never seen before. Somewhere in Europe but can't remember which.