The title pretty much says it all... (Minecraft is a game consisting of evenly-spaced cubes for terrain, like voxels)
Note: I am using C# XNA.

I am pretty sure AABB is the way to go, yet I don't know how to implement it. I admit, I'm almost looking for code, but theories/ideas are very welcome.

Important capabilities of my code: I have a function that can get a block anywhere in the world, and get a BoundingBox for that cube. Hence, I have created a BoundingBox for the player to collide with those cubes. My idea was to get the blocks around the player (maybe 4x6x4) and test against those.

The problems I have been having: Say the world is a flat plane. If I use the method of go the shortest distance out, then if the player is slightly clipped into the ground (from gravity), but even slighter into the next block over, then the player will be pushed sideways (and so cannot walk along ground). Of course, this is assuming I react to every block intersected. Another problem is knowing which direction to go (aka negative x or positive). That takes me to my final problem- Getting the amount of intersection, in the correct direction (+ or -) has been tough for me.

I hope I haven't been too hard to understand, I'm not too good at explaining things... And if this question has already been asked, I'm sorry, I looked for it... for 3 days straight.
One last thing, if someone knows exactly how minecraft does it, or has source (I know MC modders have the source, how else would they mod), please point me to it.