Check out Rough Trade's free in-store lineup here. They have The Leisure Society, Broken Records, The Drones, The Pretenders, Lets Wrestle, Bombay Cycle Club and many others...
Pure Groove also have a pretty impressive line-up of not to be missed free shows including Dengue Fever, Slow Club and Totally Enormous Extinct Dinosaurs (great name) and plenty more! Check out the full list here.
It's always worth keeping an eye out for a free Bandstand Busk as well, their website is here.
Oh and RockFeedback are hosting a FREE show at The Lexington on June 11th featuring Fol Chen, Hind Ear and Plug, more info here.
**UPDATE** Thanks to at_the_sea for leaving a comment mentioning
The FREE Daylight Music gigs at Union Chapel taking place most Saturday lunchtimes until the end of July. They also have bacon sarnies and homemade flapjacks - yum! More info on their Facebook page, here.**
Because I like they're name, here's Totally Enormous Extinct Dinosaurs dabbling in a bit of dirty techno and wearing some pretty random costumes in 'Ducks'.
Totally Enormous Extinct Dinosaurs - Ducks