Grinding the flappy bird parody also slowly gives diamonds.
Provided you have a decently strong Mon (700ish?), you can also get a few diamonds a week from the battle feature. Wait until it says nobody else is searching for a fight, then select your toughest mon. After a minute passes without finding a human competitor, you'll fight an AI with mid 600 power- likely normal or electric type so Fighting, Rock, Ground, or Ghost can get a big 25% effective power boost. DONT FIGHT IF PLAYERS ARE SEARCHING, many of them are rocking out 1300+ power mons and will punt your ass like a football.
Getting 100 rank points (20 point shift on victroy or loss, so five straight wins) gets the first ranking bonus of five diamonds, resetting weekly. at higher ranks, the AI has stronger Mons, and you're more likely be unlucky enough to run into a player.