Start with blue painters tape. You can attach it to almost any surface and it will come up easily without damaging the surface or leaving a residue.
On top of the painters tape use double stick carpet tape and then stick your floor covering on top of that. The double stick tape is too strong to stick directly to the floor, if left for any amount of time it can do permanent damage, but the painters tape protects your floor surface while giving you the stickiness you need.
I used it recently to stick burlap to the floor of this high school theatre to create a base for the "gardens" we were going to create in "Love's Labours Lost", but the friend who taught me the trick was using it to attach a masonite deck (sold in lumber stores as "hardboard") to marble, and I'm thinking about using it to cover the ugly linolium tiles in my kitchen.