Jammin' Hamster was a Flash game featured on the Pixel Chix website. It was also later added to BarbieGirls.com along with games from other Mattel sites. (This was in the site's early days, before it made its own original mini games.)
The purpose of Jammin' Hamster is to move puzzle pieces in order to create a path to get the hamsters to their rock concert before the time runs out. There are multiple levels, themed around different seasons and landscapes. It doesn't take long to beat all of them nowadays, but as a kid it took forever. There are various obstacles along the way that the hamsters can run into, like holes to fall in and puddles to slip on. There are also power ups, my favourite was the biscuit that made the hamsters run faster. The soundtrack contains only a few songs. On each level the same repetitive song plays. I guarantee that if you hear it once, you'll remember it for the rest of your life. It'll probably also get stuck in your head for days.
If you complete every level, you unlock the bonus level, which I believe featured puppies instead of hamsters. I think the music changed, too. There aren't many game play clips online for this one. If you still have a way to play Flash games, you can access Jammin' Hamster on the WayBack Machine.
The IRL Jammin' Hamster that this game was advertising to you.
Related Links
The Only Gameplay Video I Could Find