Linear programming solution in one pass with ordered variables
Probably the most famous instance in which a greedy algorithm is known to solve an LP is for the special case of the transportation problem.  Hoffman ("On simple linear programs", in Convexity, vol. 7 of Proceedings of Symposia in Pure Mathematics, pages 317-327, 1963) proved that if the cost matrix for a (maximization) transportation problem satisfies the Monge property ($c_{ij} + c_{kl} \geq c_{il} + c_{kj}$ when $1 \leq i &lt; k \leq n$, $1 \leq j &lt; l \leq n$) then an optimal solution can be found in a greedy manner like the one you describe.  

Hoffman also has a survey paper ("On greedy algorithms that succeed") from 1985 in which he discusses known cases in which a greedy algorithm gives an optimal solution to an LP.  Besides his own work cited above (about which he says, "most of the linear programming problems known [by 1963] to be susceptible to a greedy algorithm were special cases of the Monge idea"), he mentions Edmonds' linear programming interpretation of a generalization of matroids and a discussion of the case when $A$ is nonnegative, among other things.

I imagine there are more recent results, but hopefully this at least partially answers your question and gives you some ideas of where else to look.