In the past, I didn't train in the rain, but I got caught in a shoe soaking heavy rain once. And I got caught in light rains twice. The reason I didn't bike in the rain is that I wasn't waterproof, and that the bike gets messy.
Well, I just bought a great Bellweather rain jacket, a rain cap that goes underneath a helmet, rain pants, and I will buy a rain / thermal shoe booties soon. All I need now is a rain proof backpack.
Now I think, if I don't get wet with all that gear on -- I guess I can now bike and train in the rain. The only problem is handling a bike in the rain -- cornering & stopping might be slippery.
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Ok now I got a waterproof backpack -- a $3.00 special clear vinyl backpack that's tinted see through, the seams may not be waterproof but plastic vinyl clearly is. Waterproof gloves from Walmart, and I'm looking forward to getting Gordini Goretex gloves for $40. Going for a $100 Marmot Rain Jacket would be really nice, but I'll try the gear that I got on first.
I'll now only bike in the rain if the whole week is predicted to rain. If there are days of clear weather alternating with rainy days I'll only bike during the clear days. Unless I need to go somewhere only reachable by bike during the rain then I'll bike.
Rain is bad for your bike because different metals will corrode on each other. But I'll still use both my trainer bike and top of the line race bike during rainy days. I hate cleaning up after a ride, and I fear riding in 33 - 40 F rainy weather. It makes me wonder how bike messengers do this all the time.