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.
Platypus:
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 ppa:chromium-daily/stable (How to add an PPA) and add --enable-speech-input to the end of the launcher.

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.

There is a decent open source speech to text engine available:
Note that it takes a bit of configuration, which is less polished than Dragon Naturally Speaking's configuration.
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1FreeSpeech appears not to be maintained. It hasn't offered a ppa since Ubuntu 10. – JellicleCat Feb 20 '15 at 18:57
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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
