Question: I want to be able to grab multiple media clips from my project window all at once, then drag and drop them to a timeline stacked on top of each other as opposed to side by side. Each track would take its own track in the timeline.

Sort of like keyframe assisting in after effects.

Is there a way for this to be done, maybe by holding down a particular keyboard command?

Answer: 
I dont believe there is any native built in way to do this. The easiest way would likely to build a simple keyboard mouse macro which can be run to repeat the action of selecting the next clip down, and adding to a new track at the start of the sequence.
Adobe Premiere Pro CC 2015 has a thing called a Multi-Camera Source Sequence. This is kind of a combination of both a multicam clip and a sequence. I believe it is enough of a sequence that it can behave like a sequence, meaning that if you were to composite the layers together instead of switching between them, you'd get the expected result. Such sequences can be created by selecting a number of Project items, as you wish to do. When you create such a sequence, choose "In" points to sync everything, which means everything starts with each clip's first frame. For more about this new feature, read the [relevant section](https://helpx.adobe.com/premiere-pro/using/create-multi-camera-source-sequence.html#UsetheMulticameraSourceSequencedialogbox) of the Adobe manual.