Close precision cleaning, also called detailing, is required here. This is monks' work, requiring patience of Job and attention to detail.
Wet cotton buds with isopropanol and wipe the board clean. When working atop small electronic components use more of a dabbing motion than a wiping motion to avoid damaging delicate components and to prevent threads of cotton being left behind on the board. Use a pair of tweezers to remove sticky clumps and trapped threads.
When doing this I put on a CD of soothing music. J.S. Bach's Goldberg Variations played on the piano by Murray Parahia is most suitable for such occasions, and my choice for this work. Be prepared to spend an hour per side of the board.