I was there for one of the coups, and seeing pics of tanks rolling through BKK on the telly concerned me at the time, but I was reassured by speaking to people. Expats and locals alike weren't concerned in the slightest. As MrNaz says, they are a regular occurrence. Just don't be curious, stay away from the problem areas, and keep your eyes peeled on the news reports as to the status of the airports. My biggest concern in the coup I was there for was that the airport would be barricaded and wouldn't be able to get my flight home. Luckily that wasn't the case for me.
Here's the latest from the BBC... feel free to add new developments to this thread. I will temporarily make this a sticky as I'm sure other tourists will have similar concerns as to the situation.
BBC News - Thai PM rejects protest ultimatum