BookController.cs

using System.Collections.Generic;
using System.Web.Http;
using Umbraco.Web.WebApi;

namespace UmbracoTestApiAttributes.Controllers
{
    [RoutePrefix("api/books")]
    public class BookController : UmbracoApiController
    {
        [HttpGet]
        [Route("")]
        public IEnumerable<string> Index()
        {
            return new List<string> { "Return of the King", "Valour" };
        }

        [HttpGet]
        [Route("details/{name}")]
        public string Details(string name)
        {
            return name;
        }
    }
}

UmbracoApplicationEventHandler.cs

using System.Web.Http;
using Umbraco.Core;

namespace UmbracoTestApiAttributes
{
    public class UmbracoApplicationEventHandler : IApplicationEventHandler
    {
        public void OnApplicationInitialized(UmbracoApplicationBase umbracoApplication, ApplicationContext applicationContext)
        {

        }

        public void OnApplicationStarted(UmbracoApplicationBase umbracoApplication, ApplicationContext applicationContext)
        {
            WebApiConfig.Register(GlobalConfiguration.Configuration);
        }

        public void OnApplicationStarting(UmbracoApplicationBase umbracoApplication, ApplicationContext applicationContext)
        {

        }
    }
}

WebApiConfig.cs

using System.Web.Http;

namespace UmbracoTestApiAttributes
{
    public static class WebApiConfig
    {
        public static void Register(HttpConfiguration config)
        {
            config.MapHttpAttributeRoutes();
        }
    }
}