Yes, I've been able to skip the foam on unmounted rubber stamps! Especially on good quality rubber (Purple Onion are nice, deep stamps) I don't miss the foam at all.
To mount the rubber to an acrylic block I just add a strip of tape to the block and then press on the rubber. I usually leave the same strip of tape on my block for that session of stamping and remove it when I put my blocks away. The tape is a little wider than standard scotch tape.
I'm currently out of Poster tape but I think I can use it on my We R stamp positioner as well. It's a good thing to buy at Joann's with a 20-30% off total purchase coupon because it's pricier than regular tape.
Hereis a picture of it. Note the 2nd picture, it has a paper liner. It is thin just like double sided scotch tape, but less sticky (one reviewer thought it was more like foam tape, it's not.)
To store my unmounted rubber I add a strip of regular double sided scotch tape, or removable double sided scotch tape to cardstock and put it in an Avery Elle clear stamp pocket. Way back, I stored my unmounted rubber in 3 ring binders. I started out mounting on cling cushion, eventually went to coating the back with Aleene's Tack it Over and Over and they would cling to the hard plastic storage pages. Now I'm cutting up some of those storage pages and putting them in Avery Elle pockets. Lately I don't bother prepping unmounted rubber. I'm really attracted to Sadie's method! So many stamps in a little pocket, yet you can still pull out that laminated card to see what you have! Sadie, does that mean you skip the cataloging? (oops, realizing Sadie's conversation about organizing was in the other chatty thread, it's late, I'm getting my conversations mixed up!_