The obvious route would just be to expand on the world created in the films. Partly as a way to avoid the continuity issues raised by die hard Trekkies/Trekkers/whatever, and partly as a way to directly antagonise those same fans with more sci-fi mumbo jumbo, the JJ Abrams movies used the characters from the original series, albeit with different actors, and explained it away as an alternate timeline. Old Spock even turned up, Leonard Nimoy tumbling through a wormhole and then going to live in the North Pole Of Space or something! So it's canon that the new films exist in a different world to the old TV shows and feature length efforts, freeing up the filmmakers to crib from old story lines without worrying about stepping on anyone's toes. How much that works is a matter of taste (*deep exhale* KHAAAAAAAAAAAN), but it seems likely they'd do the same with a new show. Follow the world the movie audiences are used to, fans are happy they've not mucked up the history, and then you have some Marvel Cinematic Universe/Agents Of SHIELD-style big and small screen synergy.
Tom Baker is the Comics Editor at WhatCulture! He's heard all the Doctor Who jokes, but not many about Randall and Hopkirk. He also blogs at http://communibearsilostate.wordpress.com/
