Well its difficult to understand the cause of your dissatisfaction. Before hiring any design agency a client should check its portfolio, process it follow, pricing structure (Pay for project, hourly rate what ever suits you). Even at the time of final delivery there are various tools to analyze your website.
Some of various parameters which defines quality of website are -
1. Quality of content
3. Mobile Compatibility
4. Integration of analytic tool
7. SEO ready
8.Quality of code
11. Social Media Integration
12. Other best practices e.g. broken link check, url Canonicalization, Browser Compatibility etc
13. Readiness for business
14. Good look and feel
Its the responsibility of both parties to ensure above mentioned. I can understand when a client gives their project to a design company they should serve them in most adequate manner.
Note: This post was edited by a forum moderator to remove advertising promo on 04/08/2013 at 8:28 AM PT