We only have some pillows and bench cushions which all fit nicely into one of the weather-tight vinyl deck containers. They stay outdoors all year in the box.

We bought one of those big rubbermaid deck storage containers (like a small shed but only waist high) and it's perfectly sized to lay the cushions flat or on thier sides without having to fold. Fits 4-6 cushions too. It's meant to be outdoors so it takes the weather fine, even in winter.

We have the same issue and furniture like what you describe. We have two chaise lounge cushions and 6 wicker chair cushions. I don't like to leave ours out, either, as it collect pollen and potentially bird poop! We bring ours in and put it in a spare room in a huge stack. It gets old every day and is not exactly ideal, but we're not using the room so it works. I've been considering an outdoor box as described above, but it's another expense and takes space on the deck or porch.

We have a storage shed in our backyard that is the same color as our home so it's not too bad looking. That holds all the cushions plus some outdoor tools.

I put mine at the top of the linen closet that's in my laundry room. At the end of the season, I clean them and store. Not a big deal.

Red

My husband put some shelves in the garage. The cushions live there until spring.

We rent a storage house for that purpose. Although we also have two storage houses, one which is built onto (as part of the original structure) the house and the other which we built to look like a small English cottage, but we still seem to run out of storage room, so we have rented two little storage houses that are climate-controlled.

Frontgate.com has a variety of pieces that would hold cushions including a large weatherproof trunk.

We don't have deck cushions but my daughter found some over sized zip lock type backs to keep them in and then stored them in the basement.

We also have one of those Rubbermaid storage containers but ours is at least chest high (top lifts up and 2 doors swing out). It holds the double cushions of 6 chairs, a chaise lounge, and 4 folding chairs (all match).

We've used it for many years (7 or 8 maybe??) and it's held up well and protected the cushions, etc. in all kinds of weather (winter snows, spring/fall rains, summer thunderstorms, tail ends of hurricanes). It even stayed closed the couple of times we've had near-hurricane force winds and the container itself was pushed around the deck (railings kept it on the deck)...cushions, etc. stayed nice and dry inside the container!

We have a smaller one on the front porch that holds the double cushions for 4 chairs that also has works fine. Interestingly, we have had a couple of wasp or hornet nests in the one on the porch, but never in the one on the deck.

What if you tried some of the plastic covers they sell for outdoor furniture? That way you could just pop the covers on when you know it'll rain or if you just want to keep the chairs and cushions free of bird poop. I have covers for my outdoor chaise and 2 chairs, and at the end of the season I schlep the cushions inside my finished basement. If you use the chair covers, I would definitely bring the cushions in at the end of the season and try to freshen them up with the carpet stuff that you vacuum up.
I also have one of the plastic chest things, but it's not big enough for the cushions, and I'm not at all sure it's completely waterproof (I think I've seen some dirt in there, which means it's not seal-tight).