While Liviu Costea's answer is correct, it still took me quite some time to figure out how it is actually done. So here is my step-by-step guide starting from a new ASP.NET 4.5.2 MVC project. This guide includes client-side package management using bower but does not (yet) cover bundling/grunt/gulp.
Step 1 (Create Project)
Create a new ASP.NET 4.5.2 Project (MVC Template) with Visual Studio 2015.
Step 2 (Remove Bundling/Optimization from Project)
Step 2.1
Uninstall the following Nuget Packages:
- bootstrap
- Microsoft.jQuery.Unobstrusive.Validation
- jQuery.Validation
- jQuery
- Microsoft.AspNet.Web.Optimization
- WebGrease
- Antlr
- Modernizr
- Respond
Step 2.2
Remove App_Start\BundleConfig.cs from project.
Step 2.3
Remove
using System.Web.Optimization;
and
BundleConfig.RegisterBundles(BundleTable.Bundles);
from Global.asax.cs
Step 2.4
Remove
<add namespace="System.Web.Optimization"/>
from Views\Web.config
Step 2.5
Remove Assembly Bindings for System.Web.Optimization and WebGrease from Web.config
Step 3 (Add bower to Project)
Step 3.1
Add new package.json file to project (NPM configuration file item template)
Step 3.2
Add bower to devDependencies:
{
  "version": "1.0.0",
  "name": "ASP.NET",
  "private": true,
  "devDependencies": {
    "bower": "1.4.1"
  }
}
The bower package is automatically installed when package.json is saved.
Step 4 (Configure bower)
Step 4.1
Add new bower.json file to project (Bower Configuration file item template)
Step 4.2
Add bootstrap, jquery-validation-unobtrusive, modernizr and respond to dependencies:
{
  "name": "ASP.NET",
  "private": true,
  "dependencies": {
    "bootstrap": "*",
    "jquery-validation-unobtrusive": "*",
    "modernizr": "*",
    "respond": "*"
  }
}
These packages and their dependencies are automatically installed when bower.json is saved.
Step 5 (Modify Views\Shared\_Layout.cshtml)
Step 5.1
Replace
@Styles.Render("~/Content/css")
with
<link rel="stylesheet" href="~/wwwroot/lib/bootstrap/dist/css/bootstrap.min.css" />
<link rel="stylesheet" href="~/Content/Site.css" />
Step 5.2
Replace
@Scripts.Render("~/bundles/modernizr")
with
<script src="~/wwwroot/lib/modernizr/modernizr.js" ></script>
Step 5.3
Replace
@Scripts.Render("~/bundles/jquery")
with
<script src="~/wwwroot/lib/jquery/dist/jquery.min.js"></script>
Step 5.4
Replace
@Scripts.Render("~/bundles/bootstrap")
with
<script src="~/wwwroot/lib/bootstrap/dist/js/bootstrap.min.js"></script>
<script src="~/wwwroot/lib/respond/dest/respond.min.js"></script>
Step 6 (Modify other sources)
In all other Views replace
@Scripts.Render("~/bundles/jqueryval")
with
<script src="~/wwwroot/lib/jquery-validation/dist/jquery.validate.min.js"></script>
<script src="~/wwwroot/lib/jquery-validation-unobtrusive/jquery.validate.unobtrusive.min.js"></script>
Useful Links
Bundling & Minifying
In the comments below LavaHot recommends the Bundler & Minifier extension as a replacement for the default bundler which I remove in step 2. He also recommends this article on bundling with Gulp.