As a programmer who has spent the last three months learning Dvorak, I'm glad this topic came up and would love to share my experience.
Before learning Dvorak, I used QWERTY, but did not fully touch-type. I could type quite quickly, and without looking, but I did feel limited at times, and wanted to learn to touch-type correctly. I decided on Dvorak because I had heard good things and wanted to be forced to touch-type and break my old bad habits. To learn, I used an open source typing tutor, klavaro, and found it worked quite well.
The first week or so was painful. After a few days I had learned the position of all the keys, but typing had become a strenuous mental activity. Simple tasks such as writing an email became very difficult because I was so focused on how to type it that I could not focus on what it was I wanted to type. The other difficult part was passwords. It usually took a few tries before getting them correct, and sometimes I even had to resort to copying and pasting to make sure passwords were correct.
As others have noted, typing speed may not be as critical for programmers as it is for others. While this may be true for coding, I found the initial slowdown when IMing/emailing coworkers to be frustrating.
After around a month, these difficulties went away, and typing started to feel much more natural. I'm still not at the speed I was before, but typing feels so much betters now. Overall, I am happy I switched to Dvorak, and would definitely recommend it. I notice my speed improving everyday, and I actually enjoy and look forward to typing now. Using emacs with Dvorak has not been an issue for me. In fact, I would say it feels better now.
Another change I made that I feel has boosted my productivity is remapping caps lock to ctrl. Caps lock is in a great location on the keyboard but performs a somewhat rarely used function. I would highly recommend remapping caps lock to something more useful.