I'm taking educated guesses here, but I could be wrong as I am not a website developer. So, please take my suggestions lightly.
After checking your google cached version of your website I saw two things.
1] Your old links were in the format of:
The "?p=" would have me believe that your web page was built to be dynamic utilizing PHP. Perhaps your website was created to be operated with PHP and MySQL.
Perhaps since you have just hosted your web page with dreamhost you overlooked having to setup a database?
2] The second thing that I did notice was that your web page was at one point in time created by a company called, "chrysb development".
They confirmed my suspicion that they utilized PHP and MySQL to develop your website.
Up and Running
PHP, MySQL, Photoshop, HTML"
as referenced on
If you had a recent & positive relationship with chrysb development I would take a chance and ask them for a quick explanation on how to get your web page to operate properly with another web hosting company.
Seeing as how they utilize you as an example company which they created web page design services for - I would assume that they would be eager to get your web page working since on their website when you click on your company through their portfolio the link obviously does not work.
Like someone else mentioned it may have something to do with .php pages, etc.
If you still happen to have your web page files on another company's web hosting servers check out the organization of the files. What files are in what directory, etc. Check to see if something was placed in the wrong directory.
Hope this helps you out.
Good luck and please let us know if you figured out the problem.