First, I'd take a look at the error log for your server right after you tried to load a page with an image that doesn't appear. If the error is a 404, file does not exist, for one or more of the images on the page, I'd pick one image to work with, then in the back end of WordPress on the new host, go Media -> Library, and click Edit for the image. Take a look at the File URL entry and compare it to the actual path to the image file on the new server. Chances are the path to the image in the database table "wp_posts" in the "guid" column is the one for the old host. You can either move the files to the correct location on the new server, or you can run a simple SQL query in phpMyAdmin to update all the entries to the correct path to solve the problem.
If, however, the error says you don't have permission to view the image, then the permissions for the folders/file are probably incorrect on the new server. Most web hosts will run a script from a shell to change all the folder/file permissions to the correct values for you in one pass.