I.C.E. addressed this by setting their modules mainly in early part of the Third Age, most often around 1640. At this point the Shire barely exists, so your chances of bumping into (or bumping off) Frodo and Samwise are limited.
Cubicle 7 seem to have gone for the aftermath of the Battle of the Five Armies - upheaval, and familiar names in the background, but still 70+ years clear of the Ring quest.
I thought I'd go back to first principles and see what starting date would suit my purposes best. My guidelines are these -
- Some time in the Third Age. Second Age (and earlier) is very different - more high fantasy than gritty
- Set principally in Eriador*
- During or after a significant event, so that Middle Earth's history is very present rather than just a backdrop. Also, as Noisms explained, times of societal change are good settings.
1409: Invasion of Arnor
1432 - 1448: Kin-strife (but see (2) above)
1636: Fall of Cardolan
1974: End of the North-kingdom
2510: Rohirrim settle in Calenardhon
2770: Erebor destroyed by Smaug
2941: Battle of the Five Armies
Of these, the fall of Cardolan strikes me as the most interesting, hopefully not because of the deep roots I.C.E. placed in my gaming conscience. The North-kingdom is still in existance, but on the wane, so you have interesting bits of Middle Earth's history that are still current (e.g. Fornost and the Seeing Stones) while retaining that sense of the Elves and Dunedain being in decline. You have an obvious source of conflict (and adventure) between Arthedain and Angmar and more of the sort of support structure that adventurers expect in terms of safe areas and friendly forces than is the case after 1974. Speaking of which post-1974 is probably my second favourite in terms of settings, with a bit of a post-apocalyptic struggle for the remaining settlements not destroyed by the fall of the North-kingdom. I'm not sure though that it's the sort of campaign I'm particularly good at running.
* Why Eriador?
This is clearly a key question, as this preference knocks out over half of my short list. Without meaning any disrespect to the professor, the Lord of the Rings strikes me as a deeply parochial tale. In fact, I even have a feeling that this was the intent. The Shire is "home" and everything that Frodo and company do is in defense of home, rather than being about adventurers setting off with whatever (often selfish) motivation that drives your typical RPG character. Hence the view of Gondor (or Rhovanion) should be much the same as that which a British package holiday tourist had of France or Spain a generation ago - somewhere you visit, before going home again. So it's not really appropriate to have the campaign set outside of Eriador, or more precisely for the characters to originate there.