I leave my USA T-mobile SIM at home, in a cheap Android phone (with a broken screen) and plugged in. Calls go to my regular voice mail, which I can call from overseas (using Skype or GrooveIP (Google Voice). I can call my regular number, which will forward to voice mail after I don't answer, or I can call the "backdoor T-mobile voicemail number" (+1-805-637-7243) to go directly to voice mail.

Now why not turn the phone off and then calls go right to voice mail --- TEXT MESSAGES!!!

I used to bring the SIM with me, so I would be able to see SMS messages when abroad, but then I'd have to carry two phones, or a 2 SIM phone. And I also pay roaming rates to reply to text messaged. Instead, on the Android phone at home, with my USA SIM, I have two apps running - SMSForwarding by Tony Shen and mySMS. SMSForwarding sends me emails whenever I get a missed call or a text message, and mySMS allows me to reply to any SMS I receive via any browser connection, and it comes from my phone, not from some Google Voice number. So I have full 2-way SMS capability, and also get email notification of missed calls (which would usually indicate a voice mail). All this with zero roaming charges.

I write this because it addresses this issue, however, this won't work with an iPhone sitting in the USA with your SIM, as the 2 apps I mention above don't exist on iOs. But ... all you'd have to do is by the cheapest Android phone you can find (AT&T compatible, and running Android 2, at least), download and install the 2 free apps I mentioned above and put your SIM in that before you leave the US. The cheap Android I have is one with a cracked screen that someone was selling for $30 on ebay.