This, And The Other "Build Structure With X Logs" Achievements, Were Replaced With "Place X Logs" Achievements.
Place 1000 Total Logs And The Achievement Is Yours.
1) You Can Place Down One Log.
2) Then "Mouse Click" To Put Another On Top.
3) Then Hold "C" To Pick Up The Second Log.
Then Repeat Steps 2-3 Until You Get This Achievement.
(Confirmed By Deathtroth)
Good Luck!