I have a small kitchen just it is just down the hall from the laundry room.
I would think a basket would work well to store dirty linens. What about in your pantry?

I LOVE cloth napkins and only use those unless we are having company (then I get printed pretty paper napkins).

Mine is on a diff floor too. Part of my routine cleanup routine involves a trip up the stairs (or one of my minions- I mean children).

Not that big a deal. The exercise won't hurt me :)

This was recently discussed. Perhaps you can get some ideas from it.

Here is a link that might be useful: Does anybody have a laundry basket in their kitchen (dishclothes)

A doorknob hanging questionably attractive laundry "sack".

How about a pail or basket under the kichen sink? I think Circuspeanut's use of an egg basket is great! Just tuck it into a cabinet. I use a wicker basket that I sprayed a bunch of times with poly. I take them to washer once a day.

We bought a small, collapsible, tall and very thin black nylon laundry basket at Ikea. It sits on the floor at the edge of the counter overhang when needed to hold dirty linens.

Do you have somewhere on a wall or shelf where you can put a decorative, covered box or basket? No one has to know there's dirty laundry inside. :) It doesn't have to be very big if it's just the towels and maybe an apron if you think to empty it every time you're going toward the laundry room. A whole house full of cleaning rags is another story. :)

How about a small, pretty laundry bag handing from a hook inside of a broomcloset or other cupboard?

Maybe even a storage hassock outside of the kitchen, with a laundry bag inside...

I guess I'm lucky. I have a laundry walk-in closet that opens off my kitchen and I can just wad up the towels and throw them in from across the room.

A wetbag!-commonly used in the cloth diaper community, the outside is whatever fabric you love and the inside is PUL, a waterproof material that is also washable. Mine has a zipper. When you do laundry, Just dump out the towels/washclothes/napkins and add the wetbag in the mix as well. You can have them custom made places like Etsy and Hyenacart with tab hangers to fit on a doorknob or hang it on a hook inside a cabinet. My laundry room is close, but if I had one on another level, I'd buy one in a really cute kitchen fabric.

My basement opens into the kitchen, and laundry's in the basement. We've upgraded from flinging the towels down the steps to be picked up later to hanging a grocery bag on a nail at the top of the stairs. SIL uses a lot of cloth napkins, and has a shelf of one cabinet with two basket/bins on it - one clean, one dirty.

i collect mine in a small rollout hamper in my laundry closet.

A few good ideas, and my little wicker basket was great. I'm in a bi-level, no doors to anything hidden, just the DR, LR and stairs down to entry door, or hallway to bedrooms. It's all open, no doorknobs even! No pantry either. We USED to drop them down over the stair rail where they landed on the 1st floor in front of the entry door from the garage! Not cool.
Right now I have the basket in the DR, but don't want that to be my permanent solution. . .
Under sink is low because of farm sink, but it's full! Recycling bin, DW soap, other clean up stuff.
Maybe if I can find a cute basket with a lid that's small enough, I can keep it in the dining room. We used to have an old family wooden picnic basket, but I think after 60+ years it finally just fell apart.