I'm not sure if this is the correct method of dealing with the problem, but here's what I did to get "compile on save" working on my project...
After noticing that Compile On Save worked when starting from a new HTML Application with TypeScript project, I did a comparison between that and my non-working ASP MVC project. Turns out, my ASP MVC project was missing this bit of code..
<PropertyGroup Condition="'$(Configuration)' == 'Debug'">
<TypeScriptTarget>ES3</TypeScriptTarget>
<TypeScriptRemoveComments>false</TypeScriptRemoveComments>
<TypeScriptSourceMap>true</TypeScriptSourceMap>
<TypeScriptModuleKind>AMD</TypeScriptModuleKind>
</PropertyGroup>
<PropertyGroup Condition="'$(Configuration)' == 'Release'">
<TypeScriptTarget>ES3</TypeScriptTarget>
<TypeScriptRemoveComments>true</TypeScriptRemoveComments>
<TypeScriptSourceMap>false</TypeScriptSourceMap>
<TypeScriptModuleKind>AMD</TypeScriptModuleKind>
</PropertyGroup>
<Import Project="$(VSToolsPath)\TypeScript\Microsoft.TypeScript.targets" />
Copy/pasting that into my ASP MVC project gave me Compile On Save support again. Of course, it's still not exactly what I expected. The resultant JS files are not added to the project automatically, so that's something I had to do manually. I also had to manually group the JS file with the TS file to mimic the same experience I was getting with WE2.
Hope that helps someone. Would love to hear some feedback on the solution.