I've never noticed any issues with rain riding affecting my bikes. When I do ride in rain, I usually wipe down and/or clean my drivetrain after to remove the road grime. I suppose arguably there is some more wear on the drivetrain and brake pads due to road grime adhering to them, but these are consumable parts anyways, so I'm not sure I'd say this is "bad".
A bigger concern I've seen is on wet roads that are salted in the winter. Road salt can mess up all kinds of stuff if not thoroughly rinsed off, and washing a bike can be a bit more tricky in the middle of the winter so it's tempting to just park your slushy bike in the corner for a few days after a really gross ride, and that's when things start to corrode.