Sorry man, I can't agree with washing any complex electronic in water, regardless of how dry you get it afterwards. There is no way you can guarantee you've gotten ALL water out of everything even using compressed air or a warming unit. All it would take is a very minute amount of water still in one of the DIMM or CPU sockets, or a PCI slot, or in a seam on a solid cap or something else, IDK, anything, and poof, end of story as soon as you apply voltage to it.
There's a reason they created electrical contact cleaner and it's not because they just like putting stuff in spray cans.
My advice would be to clean as much of it off and out of stuff as possible, upside down wherever there is a chance of paste going down INTO something, using isopropyl alcohol and a soft bristle brush, and then finish it off with a can of mass airflow sensor cleaner which is even gentler and faster evaporating than electrical contact cleaner. It's INTENDED for use in cleaning crap off the heated wire sensor inside a mass airflow sensor unit, which is probably a bit more delicate than a motherboard.
I sure as hell wouldn't put it in a dishwasher if for no other reason than the fact that most dishwashers, unless it is brand new and has never seen a load of dishes before, is likely to be contaminated with microscopic bits of food and grease from previous washings that have not yet been flushed or filtered out of the unit. If you've ever looked inside the bottom of a dishwasher, or torn one apart for repair, and seen the areas where crap accumulates and doesn't make it out of the machine, you'd probably never eat anything off of a dish that was washed in one again. Things aren't as clean as they appear and while the final rinse does get a good amount of that off, that crap could get stuck in a slot or socket during the washing process, in addition to the aforementioned water itself.