* The biggest one is probably just train for a very long time. Competitive neural nets for many tasks are trained on GPUs, clusters, or GPU clusters, for days or weeks.
* Using convolutional layers really helps. Roughly, convolutional layers have multiple copies of the same neuron, applied to different inputs. This results in them needing to learn much less. It also leads to them kind of concentrating the gradient on just a few neurons. Because of this, if the first few layers of a network are convolutional layers, they are much easier to train. For a long time, these were the only kind of remotely deep neural networks we could train.
(I wrote a blog post on conv nets, which you can read here: http://colah.github.io/posts/2014-07-Conv-Nets-Modular/)
* So far, Michael's book has only talked about sigmoid neurons (I think). But you can use neurons with other activation functions. They still multiply their inputs by different weights and add a bias, but instead of applying sigmoid they apply a different function. Using a different kind of neuron, ReLU neurons, tends to help a lot. Unlike sigmoid neurons, which tend to have a very small derivative, ReLU neurons have a derivative of 1 a lot of the time. I've had mixed experiences, but most people swear by them.
* Using higher learning rates for early layers may be helpful.