We've been reading comments that suggest that some landlords limit the number of holes you can put in your walls.
We've lived in plenty of rentals over the years and have never come across this particular injunction, but even so, as a matter of course we've always patched any holes we've made upon moving out. Having this basic skill in our back pockets has made us feel better about the many holes we always make in the process of moving in and getting settled: to hang art, shelving, wall clocks, and here in SF, to bolt tippy furniture to the wall for earthquake safety.
For many years and many moves, I lived in 90+year old buildings whose decor included picture rails (some people mistake them for mouldings)
I always thought it was charming to hang pictures from the picture rail on a wire. If you want to get real cute you can cover the wire with ribbon.
And you can move the pictures all over - easy to arrange and rearrange without making holes.
Picture rail hangers should be available at your neighborhood hardware store.