According to this thread on MVC4 css bundling and image references, if you define your bundle as:

bundles.Add(new StyleBundle("~/Content/css/jquery-ui/bundle")
                   .Include("~/Content/css/jquery-ui/*.css"));

Where you define the bundle on the same path as the source files that made up the bundle, the relative image paths will still work. The last part of the bundle path is really the file name for that specific bundle (i.e., /bundle can be any name you like).

This will only work if you are bundling together CSS from the same folder (which I think makes sense from a bundling perspective).

Update

As per the comment below by @Hao Kung, alternatively this may now be achieved by applying a CssRewriteUrlTransformation (Change relative URL references to CSS files when bundled).

NOTE: I have not confirmed comments regarding issues with rewriting to absolute paths within a virtual directory, so this may not work for everyone (?).

bundles.Add(new StyleBundle("~/Content/css/jquery-ui/bundle")
                   .Include("~/Content/css/jquery-ui/*.css",
                    new CssRewriteUrlTransform()));

Grinn / ThePirat solution works well.

I did not like that it new'd the Include method on bundle, and that it created temporary files in the content directory. (they ended up getting checked in, deployed, then the service wouldn't start!)

So to follow the design of Bundling, I elected to perform essentially the same code, but in an IBundleTransform implementation::

class StyleRelativePathTransform
    : IBundleTransform
{
    public StyleRelativePathTransform()
    {
    }

    public void Process(BundleContext context, BundleResponse response)
    {
        response.Content = String.Empty;

        Regex pattern = new Regex(@"url\s*\(\s*([""']?)([^:)]+)\1\s*\)", RegexOptions.IgnoreCase);
        // open each of the files
        foreach (FileInfo cssFileInfo in response.Files)
        {
            if (cssFileInfo.Exists)
            {
                // apply the RegEx to the file (to change relative paths)
                string contents = File.ReadAllText(cssFileInfo.FullName);
                MatchCollection matches = pattern.Matches(contents);
                // Ignore the file if no match 
                if (matches.Count > 0)
                {
                    string cssFilePath = cssFileInfo.DirectoryName;
                    string cssVirtualPath = context.HttpContext.RelativeFromAbsolutePath(cssFilePath);
                    foreach (Match match in matches)
                    {
                        // this is a path that is relative to the CSS file
                        string relativeToCSS = match.Groups[2].Value;
                        // combine the relative path to the cssAbsolute
                        string absoluteToUrl = Path.GetFullPath(Path.Combine(cssFilePath, relativeToCSS));

                        // make this server relative
                        string serverRelativeUrl = context.HttpContext.RelativeFromAbsolutePath(absoluteToUrl);

                        string quote = match.Groups[1].Value;
                        string replace = String.Format("url({0}{1}{0})", quote, serverRelativeUrl);
                        contents = contents.Replace(match.Groups[0].Value, replace);
                    }
                }
                // copy the result into the response.
                response.Content = String.Format("{0}\r\n{1}", response.Content, contents);
            }
        }
    }
}

And then wrapped this up in a Bundle Implemetation:

public class StyleImagePathBundle 
    : Bundle
{
    public StyleImagePathBundle(string virtualPath)
        : base(virtualPath)
    {
        base.Transforms.Add(new StyleRelativePathTransform());
        base.Transforms.Add(new CssMinify());
    }

    public StyleImagePathBundle(string virtualPath, string cdnPath)
        : base(virtualPath, cdnPath)
    {
        base.Transforms.Add(new StyleRelativePathTransform());
        base.Transforms.Add(new CssMinify());
    }
}

Sample Usage:

static void RegisterBundles(BundleCollection bundles)
{
...
    bundles.Add(new StyleImagePathBundle("~/bundles/Bootstrap")
            .Include(
                "~/Content/css/bootstrap.css",
                "~/Content/css/bootstrap-responsive.css",
                "~/Content/css/jquery.fancybox.css",
                "~/Content/css/style.css",
                "~/Content/css/error.css",
                "~/Content/validation.css"
            ));

Here is my extension method for RelativeFromAbsolutePath:

   public static string RelativeFromAbsolutePath(this HttpContextBase context, string path)
    {
        var request = context.Request;
        var applicationPath = request.PhysicalApplicationPath;
        var virtualDir = request.ApplicationPath;
        virtualDir = virtualDir == "/" ? virtualDir : (virtualDir + "/");
        return path.Replace(applicationPath, virtualDir).Replace(@"\", "/");
    }