Using a street centerlines shapefile, I generated a network dataset. Using two other shapefiles (a facility layer -- hospitals; and an incident layer -- a shapefile with a single point), I am using the closest facility tool in Network Analyst.
Once I compute the closest facilities from an incident, I can display the route as driving directions from the incident to the facility. But what we need is, instead of driving directions in the form of turn-by-turn street directions, we want the route displayed as a list of nodes (i.e., a list of the OIDs from the junctions layer) that the route passes through.
So for example...instead of "Street A to Street B", we would want a list of all the nodes that are along that segment...for each part of the route. At the end, this would give a list of sequential nodes where, in theory, if you connected them you'd get the route.
Is this something that's easily do-able? Since it's not a typical request I'm not sure if it's really feasible.
You can try the Network Analyst Traversal Result add-in that Patrick Stevens posted on the Resource Center. It outputs features that are traversed during an network analysis. From there, you can use the junctions feature class, which is ordered for you. (Note that this feature class includes points that represent stops and added-cost/scaled-cost barriers in the analysis.)