Hi,
I see a lot of controversy over Babywise, and some good advice on both sides. You'll have to weigh it all and decide for yourself what works for you and your family.
First, I'd say that any organized group could be called a cult depending on how strong their particular beliefs. I don't know anything about the Ezzo's church, but the Babywise book does not espouse any religion or church, and can be followed successfully by anyone anywhere.
My story is that I love the Babywise method; I've used it with my three older children (ages 10, 8 and 6) and am now using it with my fourth (4 1/2 months old).
To start the routine, I would wake the baby up every three hours during the day (never at night), even during that newborn sleepy stage (my fourth I started right in the hospital), so they went back to sleep easily. I sometimes had to tickle their feet or use a wet washcloth on their feet to keep them awake. Good burping is essential, so they don't think they're full of milk when they have trapped air and stop feeding.
For me, "hunger cues" weren't reliable, babies suck on things for a variety of reasons. Often when I thought they were hungry, it turned out they were overtired and fell asleep feeding.
I would wake them every 3 hours if they were sleeping, but if they were awake and hungry before it was time to feed again, I would feed them early, and then work my way back onto the schedule by 15-30 minutes at a time for the next few feedings. Flexibility is a necessary ingredient, but always working back to the schedule. I post mine on the fridge so I don't have to try to recall it while holding a hungry baby!
If I found baby was consistently hungry early, I would change the schedule to make a couple 2 2/2-hour feedings, which added an extra feeding around her crankiest time of day. Again, I found it important to post the revised schedule on the fridge, so I wouldn't get confused. That really took a lot of stress off me.
If I found baby was consistently sleeping longer, I knew it was time to lengthen the schedule and drop a feeding. Babywise tells you the best times/ways to drop a feeding.
None of my children went hungry, by any means, and they're still good eaters. And they are strong and active today, with very few illnesses and no sleep issues. I watched my sister struggle with demand-feeding and co-sleeping issues, even up til adolescence, with her two daughters, and I just couldn't handle that. I love my sleep too much, I guess!
All of mine were sleeping through the night by 3 months old, and my fourth was sleeping 5-7 hours a night at 6 weeks. I started her on it right in the hospital, the earliest I've ever started the routine.
I know this is long, but I hope it helps!
Blessings,
Peg