I basically just kept a "possibility space" for each empty cell (containing a list of all possible values it could be), started with the one with the smallest size > 0, tried a value as 'true', removed that possible value from all related cells (same row, column, 3x3), recursed. If an empty cell had a possibility space of 0, it would pop back off the stack, removing that prior 'possibility' and resume from there. If I ever got back to an empty stack, there was no solution.
I threw a few dozen puzzles at it, trying to pull ones that were hard both for humans and, ostensibly, for machines, but I never saw it take more than a couple seconds, and that in the most degenerate "had to try every possible permutation" cases. Hence why I really wish I'd kept the code to throw that one at.
I knew that there could be degenerate cases like that that would cause it to take O(n!) time, but the ones that I ran into still didn't take -that- long, which was what surprised me.