Open Table's on-line restaurant reservation system.
Let's say you're looking for a 7:00 PM seating - right in the reservation sweet spot. Using Open Table you will see that,
just as you suspected, most Kansas City restaurants are booked up. However, since Open Table displays real-time table availability you could luck out and snag a table that was recently released by another diner (plans change, girlfriend broke up, whatever...). Just hit your browser's refresh button on Open Table's search results page and you will have an up to the minute view of table availability. If you're sitting at a computer all day anyway...
It's a long-shot but it can work. In a quick test this morning a 7:00 PM seating at The Rieger Hotel & Exchange (one of my favorites) became available for a few short minutes.
And of course there's always the phone. Most restaurants don't include their entire inventory of tables on Open Table. Always worth a call to the place you have in mind.