In Ruby, Sapphire, and Emerald, a Leftovers can be found in the basement of the S. S. Tidal in a trash can.
In Emerald, a Leftovers can be found as an Item Ball on the ground in the Battle Frontier. From what I understand, Leftovers have a 5% chance of appearing on rounds 2, 6, 10, 14, 18, 22, 26, 30, 34, and 38. (The chart is here if you want to look at it for yourself.)
In Emerald, Leftovers can be picked up by Pokemon with the ability Pickup when walking around. Pokemon on levels 81-100 have a 1% chance of picking up a Leftovers.
In FireRed and LeafGreen, Leftovers can be found when the Itemfinder is used on the exact spot that Snorlax was sleeping on Routes 12 and 16. If you do not use the Itemfinder on the exact same spots that the Snorlaxes were, you will not get a signal for an item being nearby.
I've put some images below that were on the pages for Routes 12 and 16 on Bulbapedia that show where the Snorlaxes were located.
Hope this helps!