I've been staring at this problem for a few days now. I rigged up this graphic to help me visualise the issue:

(from the graph, we know that the line intersects [1, 1], [1, 2], [2, 2], [2, 3], ending in [3,3])

I want to step along the line to each grid space and check to see if the material of the grid space is solid. I feel like I already know the math involved, but I haven't been able to string it together yet. I'm using this to test line of sight and eliminate nodes after a path is found via my pathfinding algorithms - my agents cant see through a solid block, therefore they cant move through one, therefore the node is not eliminated from the path because it is required to navigate a corner.

So, I need an algorithm that will step along the line to each grid space that it intersects. Any ideas?

I've been thinking along the lines of gradients, using floor and ceiling to isolate / determine spaces, or simply stepping along the line's direction vector by a fixed amount.