I beat Chess39 on my second try on the hardest setting, using a wall of all pawns (27 pawns and 4 bishops.) Keep advancing your rear pawns whenever there's space so that no pawn is ever unprotected. Eventually the pawns will advance far enough that the opponent has no room and you'll start getting traps and forks.

The key is to make sure you never leave a defensive opening - you have to watch out for two enemy pieces attacking a pawn that's defended only once. The computer/opponent will sacrifice the first piece to get the second to break through behind the pawn ranks, and it can often demolish all the pawns from there or checkmate your cramped king.