There are thicker plastic sheets with magnets dotted on some edges to seal the sheet shut. You could probably screw a hinge and arm off the top of your cabinets so the sheets open like doors?

How good it works and how nice the result is you will have to research yourself, that's all i can think of as an alternative to glass.

Have you seen the types of ikea glass doors they have? You could have several short doors instead of full length doors too.