Are you hearing just a 'click' when you try it or is it completely quiet. If you are NOT hearing any click at all, please ignore this post and troubleshoot per Berts procedure. If you are getting a click, then look into installing one of these:
http://www.greenpartstore.com/John-Deere-Starting-Improvement-Relay-Kit-AM107421.html
One way to test is to measure the voltage where the purple wire connects to the starter assy when you are trying to start it(with the black meter lead grounded of course) and see what you are getting. If nothing, then procede per Berts. If you hear it click, take a small jumper wire and connect that purple terminal to where the + batt lead connects on the bottom of the starter and if it turns over, then you know you likely have a voltage loss situation. I think(and I'm guessing JD does too) it's more cost effective to install one of these relays than to replace every switch in the circuit.
I'm speaking with personal experience. I had the click problem with my JD 170 with a 14HP Kaw engine. I can bore you with the details if you'd like, but suffice it to say, this new relay basically minimizes any losses in the wiring/switches etc that lead up to the integral solenoid. If yours doesn't have a Kaw engine, then it likely has a different solenoid setup and this won't apply. I did this to mine a few seasons ago and it has started perfectly every time since without a single "click".
The existence of this kit(IMHO!) acknowledges that the Kaw starter/solenoid cannot tolerate much in the way of voltage drop in the line that ends up as the purple wire connected to the starter assy that activates the solenoid. Since this new relay coil draws very little current compared to the solenoid coil, it pulls in very easily and connects the local battery voltage on the starter terminal to the solenoid terminal where the purple wire would go with very short lengths of wire. As the ign switch and the other safety switch(es) in the circuit wear and connections aren't pristine as new, the relay will operate more reliably.
Something to consider.....