Paper towels..
When it gets dirty, you just take the old ones out and put some new ones in..
It's very easy.
Also what makes it even easier is if you get some low, low plant stands to place the plants in that way your plants wont get root rot (stinky!!) and when you clean you don't have to lift anything up if the plants are on the stand.
You can also use those plastic shelves that you put in the cupboards as plant stands but the lowest height i've found them to be is 5 inches which is too high for my 36 inch cage.
It's easier to clean & put paper towels if everything is lifted a little off the cage floor.
And if you plan to build your cage, you wanna keep drainage in mind