You can refer to traffic flowing in a particular cardinal direction with the suffix -bound:
- I am standing on the westbound side of Beacon Street at Mass Ave.
- An accident has blocked three lanes of the southbound Santa Ana Freeway
It is far less common to use -bound with intermediate directions, but it's also unnecessary. If you're standing on the northeast-bound side of a road that runs northeast-to-southwest, you could equally accurately describe yourself as standing on the northbound or the eastbound side. Most roads tend to run more of one way than the other, anyway.
In some cases, one can also refer to inbound and outbound traffic, where inbound is toward a CBD or other central place. I can't bear the traffic on inbound I-66 in the mornings, but the crowding on the inbound Orange Line is even worse. But obviously, this does not work for travel perpendicular to the inbound-outbound spoke, nor does everyone necessarily agree on what is considered in or out.
Similarly, I sometimes use -bound to indicate in the direction of a specific place, but what is acceptable is idiosyncratic— I can stand on the Virginia-bound side of the Key Bridge in Washington, but to say the Arlington-bound side would be weird, even though it is more accurate. On the other hand, saying you're on the Manhattan-bound side of the Brooklyn Bridge is fine.
With trains and buses, -bound is used in some places to indicate direction as well. This is the Brooklyn-bound C Train, making all local stops; Train 67, the Boston-bound Northeast Regional, is arriving on Track 16; Attention customers: an outbound train, from the Loop, will be arriving shortly. But -bound is not itself necessary when travel is only possible in one direction from a particular platform. I'm at Dupont Circle, standing on the Shady Grove side.
Naturally, which side any of the above refers to will differ based on the rule of the road in the area. In my first example, since people drive on the right in Boston, the speaker is standing on the north side of the street. Were it in Bermuda, it would be on the south.