I've been thinking about this as well! We use the plastic grocery bags left over from our shopping and scoop our 4 boxes twice a day so it takes 2 bags per day. We also keep those 'full' bags in a big plastic container with a lid until it's full and then dump those bags into a garbage bag every few days.
I had been thinking about using paper grocery bags and then disposing them into the plastic box and then a few days later into a garbage bag.
But a few of our local grocers are planning on switching their plastic bags over to a new bag that looks pretty much the same as the regular plastic but is supposed to be environmentally friendly and is biodegradable.