The MyFlightbook app does it. At least, the Android version does, and I assume the iPhone version works very similarly.
For Android, go to the Options tab and make sure the "Autodetect takeoffs and Landings" box is checked. Make sure you're happy with the takeoff detection speed (70 kts by default, I believe).
You might also want to check the "Record flight path" box while you're here, if that's something that appeals to you. Note this uses your GPS sensor, so might consume battery life quicker in flight.
Then, go to the "New" tab. When you're ready, click the "Tap for now" button next to the "Engine Start" field. This is what tells MyFlightbook to start monitoring your speed for autotakeoff detection.
Go fly your flight, and after you shut down your engine, bring up MyFlightbook again. You should see the "Flight Start" and "Flight End" times have been filled in, and probably your departure and arrival airports, and any intermediate stops, too (though I sometimes have to correct this when there are two airports very close together).
Click the "Tap for now" button next to "Engine Stop" to stop recording for this flight (otherwise, it'll keep tracking you home, and to work the next day, etc....).
A very nice feature!
