A new guide (about 3 days old as I write this) suggests that the most efficient way to grind out bright dust is to run specific Crucible, Gambit, and Strike bounties. Old methods involving Bright Engrams are no longer viable since the Shadowkeep launch.
You can expect to earn around 200 per hour with the most efficient running possible.
A full data-mined list of the relevant bounties follows, copied from the guide:
Crucible
24-Hour Bounties
Hunter Slayer: Defeat 5 Hunter opponents.
Super Slayer: Defeat 5 enemies with super final blows.
Titan Slayer: Defeat 5 Titan opponents.
Warlock Slayer: Defeat 5 Warlock opponents.
Arc Slayer: Defeat 10 enemies with Arc final blows.
Auto Rifle: Slayer Defeat 10 enemies with Auto Rifle final blows.
Grenade Slayer: Defeat 10 enemies with grenade ability final blows.
Melee Slayer: Defeat 10 enemies with melee final blows.
SMG Slayer: Defeat 10 enemies with Submachine Gun final blows.
Void Slayer: Defeat 10 enemies with Void final blows.
Weekly Bounties
Combat Drill: Earn 100 points by completing Crucible matches. Defeating opponents and assisting or being assisted by your teammates grant the most efficient progress.
Quantum Mechanic: Earn 100 points by completing Crucible matches and defeat opponents with Void damage. Defeating opponents with Void Supers and Void Power weapons grants the most efficient progress.
Ride the Lightning: Earn 100 points by completing Crucible matches and defeat opponents with Arc damage. Defeating opponents with Arc Supers and Arc Power weapons grants the most efficient progress.
Step Into the Light: Earn 100 points by completing Crucible matches. Casting your Super, defeating opponents with your Super, and shutting down opponents in their Super grant the most efficient progress.
Thrill of Victory: Earn 100 points by completing Crucible matches. Wins grant the most efficient progress.
Gambit
24-Hour Bounties
Blocker Gamble: Send 5 Blockers to your opponents.
Guardian Gamble: Defeat 5 opposing guardians.
Super Gamble: Defeat 15 enemies with Super final blows.
Arc Gamble: Defeat 25 enemies with Arc damage final blows.
Auto Rifle Gamble: Defeat 25 enemies with Auto Rifle final blows.
Grenade Gamble: Defeat 25 enemies with Auto Rifle final blows.
Melee Gamble: Defeat 25 enemies with melee final blows.
Mote Gamble: Collect and bank 25 Motes.
SMG Gamble: Defeat 25 enemies with Submachine Gun final blows.
Void Gamble: Defeat 25 enemies with Void damage final blows.
Weekly Bounties
Do It: Earn 5 points. Wins add 3, losses add 2.
Might of the Traveler: Earn 100 points by casting Supers, defeating enemies with Supers, and creating Orbs of Light.
On the Grind: Earn 250 points by defeating enemies, defeating Guardians, and banking Motes.
Strikes
24-Hour Bounties
Finishing Striker: In strikes, defeat 5 enemies with finisher final blows.
Arc Striker: In strikes, defeat 25 enemies with Arc damage.
Auto Rifle Striker: In strikes, defeat 25 enemies with Auto Rifle final blows.
Grenade Striker: In strikes, defeat 25 enemies with grenade ability final blows.
Hive Striker: In strikes, defeat 25 Hive enemies.
Melee Strike In strikes, defeat 25 enemies with melee final blows.
SMG Striker: In strikes, defeat 25 enemies with Submachine Gun final blows.
Vex Striker: In strikes, defeat 25 Vex enemies.
Void Striker: In strikes, defeat 25 enemies with Void damage.
Weekly Bounties
Break The Machine: Complete 3 strikes and defeat at least 1 Vex strike boss.
End The Ritual: Complete 3 strikes and defeat at least 1 Hive strike boss.
Force Projection: In at least 3 strikes, defeat 150 enemies with Void final blows.
Lightning Strikes: In at least 3 strikes, defeat 150 enemies with Arc final blows.
Thin the Horde: In at least 3 strikes, defeat 150 Hive enemies.
Wrench in the Works: In at least 3 strikes, defeat 150 Vex enemies.
