This was originally written on April 29, 2015. Several comments have been added, and tense changed when referring to the past Tribes production.
The Deaf accent is an odd thing. It isn't acquired like other accents are though mimicry of local cultural aural traits, nor is it explicitly taught via speech therapy. There isn't any explicit marker of this accent as belonging to Deaf people other than the fact that Deaf people tend to have this type of accent.
If you've ever heard more than one Deaf person speak (in English, at least - I don't know about other languages), you'll know that their voices are just as diverse as hearing people's. Pace, volume, clarity, projection, and so on all vary from person to person. Yet, by general consensus, there is a common underlying tonality that marks the voice as uniquely Deaf. Even "tonality" has the same descriptive drawbacks as "accent" - it doesn't quite encapsulate how the sounds are vocalized differently. This consensus comes mostly from people who work frequently with the Deaf - interpreters, teachers, medical professionals, and scholastic researchers. This sound is hard for me to describe for a hearing person to conceptualize, but what I can do is describe how it feels when I speak.
Keep in mind that I learned speech when I was 9 years old, and I did it out of sheer curiosity. This means my voice may be marginally different from one who was forced to learn, or one who had a teacher that emphasized different elements of speech. All you need to note is that there's an endless variety of possible factors involved in why my voice is different. Nobody forced me to learn, as my parents and teachers recognized that I was very content with Signed Exact English until I switched to the natural American Sign Language of Deaf people in North America in my later teens.
My speech therapy consisted of cuing, which was a system of gestures associated with phonetic sounds. Each phonetic sound would be constructed in components - tongue, teeth, lips, air movement, etc - and once I got it right, the cued gesture would be associated with it for easier recall. I think it's quite a clever system as it combines elements of a Deaf person's natural mode of communication with aural elements to support the learning process. It won't work for everyone, of course, but it did work for me. A notable limitation of cuing is that one can only continue to speak and sign at the same time if one uses Signed Exact English, a manually coded language which is fading from use as ASL gains more recognition and prominence in the hearing world. It's technically impossible to speak in English and sign in ASL at the same time - the two grammars are just too wildly different.
Cuing was the bulk of my speech training, and once I'd gotten most of the sounds down, we moved onto whole words and eventually sentences. Sentences are where I started to really stumble, and never quite mastered as I became tired of the lessons and felt like I had achieved enough in this area. From then till now, my speech has been sufficiently developed to work through small situations like interactions at the cashier or on the bus (my ability to lipread notwithstanding). Extended conversations, presentations, and any other meaningful moments are out of the question - I will always sign in ASL whenever possible.
When I vocalize, I do not hear myself. I do feel the vibration of the sound in my mouth, jaw, and sinuses, but this is incomparable to actually hearing one's own voice. This is why most hearing people can modulate their voices so precisely and rapidly, and also explains much of the irritating attitude Deaf people encounter when they take longer to learn a new pronunciation. Many hearing people simply don't understand that it is a completely different learning process for us. It's no less valid, but the longer time frame is frequently perceived as a negative aspect, which makes no sense to me when people so often extol the virtue of taking one's time to do something well.
While voicing words, I form the shapes and motions necessary to produce the right sounds. They are all artificially learned - I never laid in a cradle picking up on the subtle sounds from my parents. As such, I often need to consciously process each sound I make. It's almost like a high-speed teleprompter in my head - the words slide past and I speak them as soon as they come into sight. For hearing people it's usually effortless, but for me it's like a triple layer of work: know the word, identify the phonetics, make the sounds. It might even be more than triple the work because I also have to be mindful of my breath, volume, and appropriate pauses between words.
This isn't to boast that I work harder than hearing people. The important lesson here is that speech is not natural for most Deaf people. This is why it is difficult to learn, difficult to maintain, and often damaging to the Deaf person's intellectual, psychological, and social development if it is prioritized over sign language at a young age.
For the Tribes production, my speech skill had been steadily improving as I practice more and drill down to the specific sounds that gave me trouble. I love that this experience improved my voice because that benefits myself and everyone else. It doesn't mean I'm about to abandon my sign language, the prospect of which would no doubt delight some 19th century physicians (some of them still exist today!) Rather, speech is a powerful augmentation to the array of communication resources at my disposal, but remember that I'm not merely adopting a tool of hearing people.
I make it my own. I give it my unique brand. I speak not just so that people will understand me better, but so that they will hear the differences in my voice and go, "Oh yeah, that's his Deaf accent."