  We develop the GKM theory for the torus-equivariant cohomology of the affine
flag variety using the combinatorics of alcove walks. Dual to the usual GKM
setup, which depicts the orbits of the small torus action on a graph, alcove
walks take place in tessellations of Euclidean space. Walks in affine rank two
occur on triangulations of the plane, providing a more direct connection to
splines used for approximating surfaces. Alcove walks in GKM theory also need
not be minimal length, and can instead be randomly generated, giving rise to
more flexible implementation. This work reinterprets and recovers classical
results in GKM theory on the affine flag variety, generalizing them to both
non-minimal and folded alcove walks, all motivated by applications to splines.
