Scan the Support Community sites and you will find: Apple TV losing connection is prevalent. The many postings point out the problem occurs with:-
1 Apple TV1,2and 3;
2 with different brands of router including Airport Extreme;
3 with current and past versions of iTunes;
4 after varying periods of working well.....minutes, hours, days;
5 losing connection with Home Sharing and losing Airplay;
6 whether the router and Apple TV are in same room or separated by distance;
7 after trying innumerable posted 'fixes';
8 after exchanging for a new Apple TV;
9 with all the recent operating systems -Mountain Lion, Lion, Snow Leopard, Leopard and Lion.
10 when it loses wifi, the ATV will work immediately if you plug in the ethernet cable.....Steve Jobs called the ATV a hobby, but Apple products are supposed to be better than this.
I suspect frustrated readers just give up trying, rather than find a fix, which is why I invite readers to please don't post a fix until you check it works for at least a week.