I've been hearing different things depending on whom I talk to about rotating tires. Some mechanics say to rotate front to rear and never cross the tires from left to right. Others say it doesn't matter.
As you said, it depends on whom you talk to. Now-a-days, the different tire manufacturers have different philosophies. For example: Michelin and Goodyear say left/right crossing is a correct part of normal tire rotation. On the other hand, Pirelli may void your warranty for going side to side with their product. Don't ask me how they would prove it without a polygraph.
There are situations where rotation is limited due to directional tires or different size front and rear. In general, the most common practice is crossing the rear tires to the front and then rotating the front tires straight back on a front wheel drive car, and crossing the front tires to the rear with the rear tires going straight to the front on a rear or four-wheel drive vehicle.
The best thing to do is to consult your owner's manual, and follow the manufacturer's recommendations. As long as they coincide with the warranty requirements of the company whose tires you're riding on, you should be OK.