Originally Posted by
grumpus
Check the frame for cracks around the bottom bracket, particularly seat tube and chainstays. With the bike on its wheels, lean it away from you and press on the bottom bracket area with your foot, looking and listening for anything that shouldn't move. Repeat from the other side. It probably isn't cracked, but this could be the cause of your problem and it's a quick and easy way to check.
The OP said that he replaced the front shifter cable because it looked a little frayed. Quite a leap from that to a cracked frame.