Characterizing earthquakes using accelerometer data
Yes, you can measure seismic energy with accelerometers. Most people do.

There are three ways to measure seismic energy: by displacement $x$, velocity $\dot{x}$, or acceleration $\ddot{x}$. Loose-spring devices like geophones measure velocity; stiff-spring devices like MEMS and relative gravimeters measure acceleration: they are accelerometers. Those used in seismology and exploration geophysics are very precise.

I wrote about the various instruments used to measure seismic energy in a blog post a while back. An excerpt about seismometers:


  Seismometers [are] precision instruments not used in exploration
  seismology because they are usually quite bulky and require careful
  set-up and calibration. [Most modern models] are accelerometers, much
  like relative gravimeters, measuring ground acceleration from the
  force on a proof mass. Seismometers can detect frequencies in a very
  broad band, on the order of 0.001 Hz to 500 Hz: that's 19 octaves!


Short story: When digital seismic receivers arrived on the scene in Calgary in the early 2000s, there was a lot of skepticism. We spent a lot of time comparing them to traditional geophones. The conclusion: they were about the same. Here's one comparison by Michael Hons at CREWES, and here's another; here's yet another from one of the big manufacturers, Sercel (now CGG). Finally, here's a presentation more oriented around earthquake seismology.

Okay, but I was talking about the accelerometers in iPhones!

Oh, sorry. As @Fred points out in his answer, consumer-grade MEMS accelerometers (as you might find in a phone or latop) may not have the sensitivity or calibration that a researcher would want, but one might compensate somewhat for this with multiple measurements, e.g. an array of devices. In other words: quantity, not quality. (This is perfectly legit: we do it in exploration seismics too.)

Dashti et al. (2013) in Earthquake Spectra Journal found you could get useful measurements (e.g. for emergency response) from networks of phones with a few tricks:


The more measurements the better.
Horizontal orientation is best.
Prefer high-friction phone covers.
Don't expect to get great data for small quakes.
Treat small groups of phones as arrays.
Don't use data for phones that fell off a table!