I've only gotten caught in the rain once, but it wasn't just rain... it was a thunder storm + flood. Not good. Given that the weather in Texas can get crazy very fast, I don't bike when the forecast calls for thunderstorms (we hardly ever just get rain; it's most often thunder/lightning/rain, which is dangerous). Add that to the fact that my city is very windy and it's just downright dangerous to bike in South Texas rain. That day I was biking home and a kind person with a bike rack on her car stopped and drove me the rest of the way. Water was nearly all the way up her car's tires.
Luckily, it doesn't rain all that often here. However, I always bike in bike shorts and a breathable fast-dry long-sleeve shirt (to protect from the sun). I carry my work clothes in a bike bag (not waterproof, I discovered). And because I work in a professional environment, and it's muggy and/or hot for 9 months out of the year, I shower at work. I would stink if I didn't, and that's not great when you're dealing with a client most of the day. The work shower sucks, but it's better than nothing (negligible water pressure).