As this question has been left unanswered for a year, I thought that I might as well answer it.
The only way to start a new game without losing all of your items is to trade Pokemon holding the items that you want to keep to another game, and then trading them back, once you've restarted your game.
It'd probably be fastest and safest if you had two system and two games of the same generation. That way, the only person you have to trust to trade your items back is yourself, and that shouldn't be a problem for most people.
If you don't have two systems, but do have two games, you can put Pokemon holding the items you want to keep on the GTS and accept the trade on your other system. Obviously, the problem with this is that there's a chance that you won't be the first person to get to your trade, so somebody else could end up with the item you wanted to keep.
You can also, as Fizz said, find somebody on the internet who's both trustworthy and willing to take a bit of time to receive your items and trade them back to you.
Also, you cannot trade Z-Crystals, you can't keep the event-exclusive ones.