Welcome and thanks for the question, I think a super relevant one to a lot of folks! Sounds like you'll have to come up with unique solutions depending on the item you need to move and how far and to/from where.

If cases have table legs there may be a way to stabilize the legs by adding temporary shear material between the legs so they won't be subject to lateral forces on their own.

For the heavy 3' pedestals, if they can be tipped slightly, try blocking one side (get creative here: enough people with their feet against the bottom edge, or a piece of plywood with heavy weights on it against the edge, ...) and then pushing on the opposite top edge of the case to get it tipped just enough to slide a pair of strips of wood or plastic underneath the edge. Now you might carefully use two prybars (you can wrap the bar end with tape or attach a thin pad to its end for protection) to lift the case along the sides as close as possible to where the case meets the floor. Or use the same method of blocking one side get the other up enough to slide strips underneath. Once you have both sides up you can continue lifting and adding strips to get the case up to a height where a pallet jack can maneuver underneath. You're right about adding blocks underneath to make the effort easier next time. If the case has smooth enough sides, you might have luck with suction cups to make the initial tips happen, though be careful should they not grip well enough.

Feel free to share photos in your posts if you like, it may help for making further suggestions.