1. Linear Regression (which, admittedly, was amazing)
2. Fourier Analysis (which is linear regression on orthonormal bases of functions. it blew people's minds)
3. Perceptrons (which is linear regression but with a logistic loss. it went back to its old name of "logistic regression" once its insane cachet of biological plausibility faded)
4. Neural Networks (stack of logistic regressors. popular with people who didn't know how to filter their inputs through fixed or random nonlinearities before applying linear regression)
5. Self Organizing Maps and Recurrent Nets (which were neural nets that feed back on themselves)
6. Fractals (which is recursion. they were useful for enticing children into math classes)
7. Chaos (which is recursion that's hard to model. useful for movie plots)
8. Wavelets (which is recursive Fourier analysis, and probably still way under-used)
9. Support Vector Machines (which replaces logistic regression's smooth loss with a kink that makes it hard to use a fast optimizer. often conflated with the "kernel trick", which appealed to people who didn't want to pass their inputs through nonlinearities explicitly)
9. Deep Nets (which are bigger neural networks. the jury's out whether they work better because they're deeper, or because they're bigger and require a lot of data to train, or because they require a programmer to spend years developing a learning algorithm for each new dataset. also whether they do actually work better).
Once this Deep Net thing blows over again, my money's on Kernelized Recurrent Deep Self Organizing Maps.
(On a serious note: MNIST is considered a trivial dataset and doesn't require the heavy machinery of deep nets. linear regression on almost any random nonlinearity applied to the data (say f(x;w,t)=cos(w'x+t) with w~N(0,I) and t~U[0,pi2/]) will get you >98% accuracy on MNIST.)
I would say that the next bing thing is more: Realizing even more that Neural Nets is an optimization problem, and instead of using some heuristics, wait for some Russian mathematician to derive the right SGD schedule / batch solver for the problem. Then what the 1,000 of Google computers have been able to do for the cat face detector, we'll be able to do it on a smartphone chip. People have to realize that Deep Learning is a bit of a "brute force" solution for the moment (each node is a linear model). We need to derive smarter algorithms.
The next big boom in AI (ignoring some logic/rules-based research in the 70s that I don't think is very interesting from an AI perspective) occurred in the 80s, when computational power increased and researchers discovered/rediscovered neural approaches, including the obscure 1974 research on backpropagation. This led to tons of press and funding from governments who dreamed of killer AI robots and what-not. But, once again, imagination raced ahead of reality and funding dried up when said robots didn't materialize. The field didn't really die off, but funding in AI went way down, leading to another major "AI winter".
I'd say the next big era of AI is the one we're in, driven largely by applied statistics that became known as "machine learning". This has been by far the most successful era, and has probably added 100s of billions of dollars to the economy (I'd argue Google is a machine learning company, for example). I think it's also the most pragmatic era, as people in the field have really learned from the past mistakes of overpromising. In fact, when I was studying "AI" in grad school, my professors warned me to always refer to what I did as machine learning because the concept of "intelligence" was such a joke to so many in the field.