While IP is not the most complex area of law, and is definitely one of the more attractive ones for laymen to think about, I can assure you that there is a 99.99999% chance that anyone drawing legal conclusions (e.g., this or that is copyrightable) on the internet has no clue what they are talking about. The fact that I see at least three different countries represented here (and all weighing in on IP law) certainly gave me a hearty chuckle (laws aren't the same in different countries). I know it's well-meaning, but if anyone isn't sure whether or not something they are going to do is legal, ask a lawyer, not the internet.
I'll try and break it down the best I can (for America), but you shouldn't listen to me if you actually think you're going to be doing anything questionable.
A technique is not copyrightable. A trade name is not copyrightable. If they were, we wouldn't have patents and trademarks (makes sense).
They have probably registered their name as a trademark, and even if they hadn't I wouldn't go around using it.
The artist certainly owns a copyright in any drawing or doodle he has made. You automatically own a copyright in the squiggly lines you make in the margins of your notes in class. Probably in your shopping list, too.
As for techniques, those would be covered by patent law. I would be willing to bet a lot of money that "a new type of doodling" isn't going to pass muster in the Federal Circuit even if they somehow bribed the right people at the patent office. Also, patents aren't like copyrights; they aren't automatic. They would have to actually register their new doodling technique before they could protect it, and like I said I don't think that's going to happen.
So, that said, if you drew a picture that kind of looked like their pictures, you'd probably be ok.
Actually, I just looked at their website. They have apparently registered their trademark in their name, but again that would be stupid to use even if they hadn't. They have also applied for a patent for their teaching method. Notice that's not the techniques they are teaching, but the way they are teaching them.
More importantly, they even say you can sell anything you make with their techniques! (
http://www.zentangle.com/index.php?option=com_content&view=article&id=28&Itemid=29) So even if you don't believe me when I say that you probably could anyway (and like I said, you shouldn't believe me!), they don't care and aren't going to come after you. That link seems like a pretty good overview of what they are claiming, and while some of the ways they word things are a bit misleading, they don't seem to be overclaiming anything. Check it out. They appear to be a rare case of a company protecting what they have a right to diligently while at the same time encouraging the wide and free dissemination of what they don't have a right to.