Is there a Ubuntu alternative for this program?
There is a whole Article on Wikipedia dedicated to the Problem.
Unfortunately, it seems there's currently no one solution that works well enough, but a massive list of projects that are underway.
You may want to rummage through the list of solutions to see if there's anything that fits your needs.
There's always the possibility to run Dragon Naturally Speaking using WINE. The Platypus Project is creating software to redirect Dragon's output to any Linux / X application, and they are very far along.
You should definitely take a look at this.
You can install Chromium 11 version from the ppa
Now visit this website and whoo! Speech-to-Text on ubuntu using html5.
I read here that if you speak slowly it will be more accurate.
To expand on Alaukik's answer, there are apps built on top of chrome speech recognition capabilities.
I personally use Dictanote ( it is very similar to Dragon ). Here is a link to chrome app : https://chrome.google.com/webstore/detail/dictanote-speech-recogniz/aomjekmpappghadlogpigifkghlmebjk
The site itself : https://dictanote.co