Breaking News!!! As of August 2018, a SmartThings-connected Contact or Motion Sensor can now trigger an Amazon echo routine if you live in the US, Canada, or the UK !!!
Amazon has now added the ability for some sensors to trigger an Amazon echo routine (not a smartthings routine), which does let you trigger other smart home devices which are not connected to SmartThings without having to speak an echo command! Or you can set it up so that your echo devices will speak a customized phrase, play a doorbell chime, or various other sound effects. 
And even better, that contact sensor can be a virtual contact sensor. And although it’s a bit tricky, a community member has created an edge driver that will create BOTH a virtual switch and a virtual sensor and keep the pair in synch. So you can set up the logic on the SmartThings side however you want, then turning the switch on will mark the sensor as “opened.“ So you can just turn the switch on and off with a SmartThings routine or other automation. 
If you have a SmartThings/Aeotec hub, you can use the following community – created edge driver to create the virtual device you will need. (if you don’t have a hub, keep reading: there are other alternatives as well.)
[ST Edge] vEdge Creator: a virtual device generator for end users
The setup using a physical contact sensor is very easy. The set up using a virtual contact sensor is more work, but may well be worth it.
This now works for US, Canada, and UK accounts. It is likely that more countries will be added in the future, just check your Alexa app to see what’s possible.
Here’s the Alexa article on creating a routine.
https://www.amazon.com/gp/help/customer/display.html?nodeId=G2PYLKJN3XVZ55EQ
Routines are in an if/then format, only Amazon calls it
When this happens/action
For “When this happens,” choose “Smart Home” and then pick the sensor (physical or virtual) you want to trigger the routine.
For “Add action,” choose “Smart Home” if you want to have Alexa control a device.
Choose “Sounds” if you want to have Alexa Play a sound effect like a doorbell chime or dogs barking.
Choose “announcement” if you want to have any or all of your echo devices speak a custom announcement.
Choose “Skills” if you want to have Alexa activate one of your skills without having to speak to it.
(Note: prior to March 2023, you could create a single virtual device which was both a contact sensor and a switch, and you will see much discussion of this method in older threads. However, that no longer works, so you will now need to use the synchronized dual devices method. )
December 2020 Update
If you live in the United States, Amazon has now added the ability for a “custom action” as the last step in a routine. This allows you to have a routine do anything that you can speak to Alexa. This is a great way to add partial Integration with devices/skills that work with Alexa but not with smartthings. However, it is still only one way: you can have smartthings as the “if“ using the virtual Sensor method and the other device/skill as the “that“, but not the other way around for most items.
https://www.amazon.com/gp/help/customer/display.html?nodeId=GLXY7RFX3L5GH2VR
December 2022 Update: If you don’t have a SmartThings/Aeotec Hub
In the old architecture you could create a combination virtual sensor/switch whether you had a hub or not. But now the synchronized dual devices are done with edge drivers, and edge drivers need a hub. So if you don’t have a hub, what do you do now?
Fortunately, since this FAQ was originally written, Amazon has added the ability to trigger an Alexa routine from a lock also.
And in December 2022, a community member created a webpage that will allow you to add a virtual lock to your SmartThings account without requiring a hub.
It won’t feel quite as intuitive, because instead of turning a switch on and off, you’ll be locking and unlocking your virtual lock. But otherwise it works pretty much the same way.
You put all of your logic for when you want to trigger the Alexa routine on the smartthings side, with the last result being that you lock your virtual lock.
And you also create an Alexa routine, which is started by your virtual lock being locked.
Here’s the thread about creating that virtual lock:
And if you have household members who would be confused by telling Alexa to lock a lock when they want to, say, turn on the fan, you could create a simple on/off virtual switch, and use a routine so that when that virtual switch turns on, your virtual lock locks. (Do that on the smartthings side, because Alexa is going to want a pin number. )
So it takes an extra set up work, but you should be able to eventually make it all happen without having to buy an additional device.
Alternatively you can do this without using virtual devices at all. Use a smart plug which visible to SmartThings and can be used as the “if this happens” trigger in an Alexa routine. At the time of this writing there were only a few of these, including the Hue smart plug if connected to a Hue hub. This will not require a custom edge driver and is quick and easy to use, but does cost about $35.
Using a physical button to trigger Alexa routines
We should also mention that as of November 10, 2018, Amazon echo buttons (two for $20, about the size of a deck of cards) can also trigger an echo routine. So you can press the button and have the echo routine turn on a virtual switch which then triggers pretty much anything you want in SmartThings. 
The button won’t be visible to smartthings, but it’s a nice alternative means of control if you already have an echo in that room.
Also, as of November 2019, they have also added some sound effects options, including doorbell chimes, a dog barking, and sirens. Thanks to @fido for pointing that out in another thread!
And as of March 2021, Flic buttons are another physical button that can be used to trigger an Amazon routine.
Flic buttons now trigger Alexa Routines (March 2021)
And as of December 2022, some switches and buttons connected to a hue bridge will be recognized as “remotes” in an Alexa routine and can be used as a trigger, along with some Hue-connected bulbs and plugs.
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