The "BIG" Games for the teams you've listed are as follows:
Beechcroft: Columbus St. Charles (D2) 8/29
Rogers: Toldeo St. John's (D1) 10/3
Toldeo Whitmer (D1) 10/17
Portsmouth: Portsmouth West (D4) 8/29
Ironton (D4) 9/12
Gallia Academy (D3) 10/10
Columbus St. Charles (D3) 10/24
Dover: Uniontown Lake (D2) 9/12
New Philadelphia (D2) 10/24
Valley View: West Milton (D4) 9/26
Bellbrook (D3) 10/17
Dayton Oakwood (D4) 10/24
The Rogers/St. John's game should be a pretty good game. Portsmouth has a natural geographic rivalry with Portsmouth West however, their biggest game of the year (and the biggest regular season game in southern Ohio) is Portsmouth/Ironton. That game is at Ironton this year and the home team usually has the edge, historically. Both games I listed for Dover are big rivalry games between the respective schools, especially New Philadelphia. If I had to choose though I would go to the Dover/Lake game. I don't know much about Valley View other than they're a very good team who post impressive marks (9-1, 10-0) consistently. After looking at their schedule I can see why they suceed so well in the early part of the season, they have a tough final 4 weeks though. I would put my money on the Valley View/Bellbrook being the game to go see. Late season matchup that could have tremoundous impact on the computer rankings. If you're going to go watch potential recruits why not watch them when the pressure is on? If you need any more info let me know.
-Turf Man