So I began using Android Time Card on my phone, and Dropbox,
so far it's a simple and efficient way of tracking hours.
Android Time Card allows you to track and save time worked and send the data (as a .csv file) to a variety of places, i.e. you can e-mail it to yourself (or your employer) or share it in a variety of ways.
I've found that with Dropbox installed you can send the .csv file to your Dropbox - works great if you have a "timesheets" folder.
What's nice is that by exporting it to Dropbox you have access to it on any device that has Dropbox synced (i.e. my work computer, my laptop, my phone, etc.).